The first method almost guarantees that his family would have to fight with the titan shifter clan, with a strong possibility that they'd lose.  I mean, Frida lost against Grisha, and his everyday demeanor is that of a normal man wrapped in a marshmallow and deception. The second option allows Rod not only to stay in the relative peace and quiet behind the walls, but probably stay alive, quite likely keep his crown, and preserve the philosophy of the First King (hereinafter "1K"), who for some reason felt it necessary not to destroy the titans outside the wall and keep humanity relatively content and clueless inside the walls.

Consider this: if you don't erase too long a period of time (say, an entire growing season, or any time where the leaves have fallen off the trees, or a period of time with able markers that even a person without a calendar would notice), you could probably get away with forcing everyone inside the walls to forget that a 55m-tall Visible Gumby and a 15m-tall Rob Gronkowski lookalike broke through Shiganshina, was compelled or beaten back, and that everyone not responsible for food production had to go rebuild the city for no pay.  Hell, get Shiganshina rebuilt in a year to the day and you could hypothetically pretend like nothing ever happened.  It just becomes a collective nightmare that everybody eventually forgets.

Rod Reiss is many things, among them a pretty skilled dead-eyed liar and an inveterate coward.  So if this is an option, I bet he knows it.  And I bet he knows it because something like it has probably happened before (I'll have at least one theory for Friday about this, and I promise I'll actually post that one before midnight), and quite likely more than once.  But how would we know that?  Well, 1K was nice enough to give us a control group: the minority noble families, including the still-nameless Asians and the Ackermans.

First, let's get the collective assumptions about the Ackermans out of the way:

1) They tend to run from "taller than average" to "Tall Drink of Water" in height (obvious exception: Levi), with an athletic build scaled to their height

2) They age ridiculously well (case in point: Levi)

3) They heal like they're the actual descendents of Wolverine (Free Crack Week Thesis: The Ackermans Are Actual Decendents of Wolverine, But, Like, Probably Hugh Jackman Wolverine Because Most of Them Are Not Particularly Short)

Gramps Ackerman (hereinafter, "Ackergramps") tells Kenny in Chapter 65 that he was one of the first generation born inside the walls.  According to the timeline given at the 104th Cadets Corps's graduation, the titans first appeared and began to gnosh on humanity in 743 (107 years before 850). This means the earliest Ackergramps could have been born was 743/44.  Kuchel's kid that Kenny references is likely Levi, and if Levi is in his early 30s in 850, that puts the Kenny/Ackergramps conversation happening somewhere in the 815-820 range.  If he was born immediately upon getting inside the wall, his would make Ackergramps (hereinafter hereinafter "AG" because "Ackergramps" takes way longer to type than I ever imagined) somewhere between 71 and 77.

That's certainly a reasonably ripe old age for someone living in a civilization with tech/aesthetic that falls someewhere between Pieter Brugels paintings and Steampunk WWI.  But given the definitey-not-Wolverine healing factor, dying in your 70s seems somewhat unbecoming an Ackerman. Given how it seems to keep them baby-faced into old age (for god's sake look at early series Levi he looks like a 12 year old with chronic hay fever and/or insomnia, and Chapter 65 Kenny has the physique of a tall guy who is old enough to have filled in from a gangly-tall teenager phase but who stole a young man's face, House of Black and White style...an Ackerman is No One), I don't think it would be out of the question for Ackergramps to be dying at an age that most people would find preposterous, and despite “average” ages from A Long Time Ago being ridiculously low, living into the 70s isn’t exactly preposterous.

(Seriously, you hear about how the "average" age of someone in, say, the middle ages was around 30 and you think everyone drops dead at 30.  No, the extremely high infant mortality rate and the fact that we used to put kids to work with farm equipment and early Industrial Revolution equipment and animals that like to kick small people who act like jerks around them meant that a lot of people died really young in the past and skewed the average down.  If you made it to adulthood, odds were pretty good you'd live long and prosper.  If there wasn't a war.) 

Additionally, consider that AG is getting information about their family's service to 1K entirely secondhand.  If he were born almost immediately when 1K's reign began, you'd think he'd have some vague memories of his family's prosperity (unless the Ackermans almost immediately annoyed 1K which, given how the most recent 2 generations behave, is...entirely possible), of a time when they needed to flee.  It's quite likely he was born after they pissed off his royal titanship and went into hiding.  Which would make him younger than his 70s.

And I've got further reason to believe he was born well after the walls went up. (That's right, I'm still not done with this theory. Thanks for reading, btw.) And that reason is this weird kid:

When I first read Chapter 5 ("Side Story: Captain Levi"), I'd initially assumed that creepy kid with the baggy Levi eyes was a young Levi, before I realized that was supposed to be the very distant past. (Free Crack Thesis: That Creepy Kid Totally Is Levi and Levi is over 107 Years Old.)  Visions of the past and visions from a distant place don't happen in this series without a character we know being present.  When someone isn't present, or when they're imagining the worst thing that could happen, those panels are usually grayed out.  The opening pages of Chapter 5 are not grayed out, so it's reasonable to assume that we're looking at something that actually happened, and given the context of this being a side story about Levi, it's also reasonable to assume that this Levi-looking kid (and the vaguely Kenny-looking adult) are distant relatives of his.  

Someone had to live long enough to tell the story of the time they were surrounded by titans, long enough for their panels not to be grayed out by heresay. At least one of those three got to Wall Maria, and I strongly suspect that their  trip involved a calm exterior, audible heartbeat, and very-much-not-dead-eyed Murder Visage of an unstoppable Goliath-slayer.  And it wasn't either of the frozen-solid adults.

Call Creepy Eyes Baby the First Ackerman!  He made it to Wall Maria with a trail of evaporating titan corpses behind him, became an immediate hit with 1K (who was none too keen on other titans) and the nobles (who probably didn't mind the idea of having someone around who could somehow murder giants, in case 1K got out of hand).  He eventually married and settled down, maybe had a few sprogs of his own before pissing off 1K, sending his family into hiding, and offering himself up as a living effigy to the Ackerclan.  And unless he was having kids as a teenager (not out of the question, but we’ll assume he was at least in his 20s), he was likely within the walls for over a decade before he had any kids of his own.  Putting Ackergramps in his 60s, which is a pretty unreasonably young age for someone to be dying of being old, especially when he’s a decendent of Bruce Willis in Unbreakable.  And considering that Kenny could well be approaching that age when he first appears in the narrative but doesn’t look like he’s been pre-emptively mummified, the "107 years" timeline we've been given seems more and more inconceivable.

(Aside: And holy crap now I want the story of Highly Resourceful Ackermom Who Didn't Have the Power of A Genetic Murder Visage and Had To Somehow Keep Her Family Safely Hidden From Her Late Husband's Enemies, Who Include Not Just The King But Aso Every Noble Family With the Financial Means To Hunt Them Down Like Injured Foxes, But For Some Reason Couldn't Tell Them Why They Were Running Before She Herself Also Died.  Because it may well have been "The Ackermans don't think we should re-wipe humanity's memory after we've cleaned up this titan attack."  And you'd best believe 1K wiped humanity's memory of the Ackermen and the unnamed Asian family once they were persona non grata.  You don't want anyone loyal to the people who speak out against you.

Oh, fascism. You so untenable.)