The real question is: which forms of investiture are permissible to use on Shabbat?
Imo:
Surgebinding is out, Stormlight Access is usually dependent on having money on hand, and far more importantly like half the surges are straight up forbidden (transformation, light manipulation, etc).
Allomancy is in (contextually): You are just eating and then using the energy you ate. In Addition none of the the allomatic abilities are straight up assur, just contextually.
Feruchemy: 50/50, tapping is clearly ok. But Filling for the purpose of using after shabbat is forbidden, BUT filling for the sake of gaining the benifit of filling on Shbbat is permissible. (E.g. Forbidden to fill cadmium/breath because there is no reason you would want to be gasping on Shabbat, but it is permissble to fill brass/heat if you are overheating).
Hemalurgy: Obviously creating a spike or spiking someone is forbidden (probably straight up, not just on shabbat), but once you are spiked, it defaults to the rules of the system in question.
Forging: Forbidden, to much of a change, even if it isn't "permanent".
Aon Dor: Literally no idea.
Awakening: Also kinda unsure.
Problem: I feel like a lot of your reasoning is more vibe-based than fully based on the laws. To be fair, I do think you're somewhat right, but let's actually look at things.
There are thirty nine primary forbidden actions during Shabbat. Outside from that, there are further prohibitions made for a variety of other reasons - preventing getting close to breaking one of the 39 or acting in un-Shabbat-like ways. Things can get vague, but we'll see.
You are assuming that using Stormlight is inherently breaking Shabbat, which itself is interesting. Yes, it's forbidden to carry money on Shabbat, but the prohibition is DeRabanan, meaning it comes from the secondary set of prohibitions. I don't fully remember the reasoning, but I think buying and selling on Shabbat is somewhere around "might cause you to write" or "too much of a weekdays' thing", and it in turn led to money becoming item used for forbidden actions. Which is about as convoluted as why musical instruments are Muktze if not more so, but nobody's asking me. Stormlight itself is energy, might be considered fire by some people? But even electricity isn't actually widely considered fire, despite what people think, so maybe not. If the use of Stormlight in itself isn't forbidden on Shabbat, spheres become no longer Kli sheMela’chto le’Issur, and thus might be allowed to carry on Shabbat.
Some of the surges, such as Division, are likely prohibited due to being prohibited actions: division is known to sometimes light fires. Cohesion and Tension seem to be creative to degrees that might make them considered building. Division is also mostly destructive, which may give it some grey area, but not much. I do agree that transformation would be a problem - that would be Nolad IMO, like ice melted or frozen on Shabbat. So that's out. Regrowth is healing, which is sort of prohibited on Shabbat if it's not to save lives? However, the original reasoning isn't too applicable here IMO, so it might be allowed regardless. Teleportation and Gravity are a bit weird - there are travel limitation on Shabbat. Gravitation in itself might not be that much of a problem, but using it to make objects fly... debatable, possibly fine.
I actually disagree with you about Illumination. People tend to see light as an extention of lighting a fire, but it's actually not that much of an essential change. Shallan tends to draw to use this power, which would be prohibited, but we know she's an oddity. Abrasion and Adhesion, the final two, are opposites - one sticks, one changes friction. I can't think of a direct Melacha related to each - maybe tying with Adhesion? Maybe tanning with Abrasion? But they might be fine.
Allomantic abilities. Well. There might be some Shechikat Samemanin issue here - shaving metal bits might be problematic. If you already have reserves, though... tin and pewter seem decent enough, maybe ‘Uvdin deḤol though - it's not very Shabbat-like to enhance your strength, and sports are prohibited on Shabbat... somewhere... on the second to third tier... I don't know. Iron and steel involve moving objects, which could be a problem if there's not Eruv. Easy to forget, moving objects more than four cubits in public spaces is prohibited.
Zinc and brass should be fine, IMO - it's happening in people's heads, so nothing creative happening in the world. Copper and Bronze are more on the ‘Uvdin deḤol area, so I'm a bit vague on them. Possibly allowed, especially if there's Pikuach Nefesh. Gold and Electrum are... well, kind of useless. I don't think there's much problem. Atium is a serious problem because it's usually associated with fighting and killing. Malatium can get an edge. But we don't care about those, do we?
Cadmium and Bendalloy, now that's interesting. Personally I think Cadmium should be prohibited because it allows you to shorten your Shabbat. Well, maybe not completely prohibited, but certainly discouraged. Aluminum and Chromium are destructive, which is probably in the same grey area as Division earlier, while Nicrosil and Duralumin... IDK. Since the whole burning thing is a metaphor, it's probably fine? But Allomancy might be completely prohibited due to that metaphor.
Feruchemy... well, I don't fully agree, again. Filling for the sake of having it later isn't allowed. Filling for the sake of the effect itself, though? I think it should be fine too. Pewter is again on a problematic area, IMO. Tin should be decently fine. Steel is probably with Pewter but allowed on certain conditions, iron is likely completely fine. Zinc gets a pass, not sure about Brass. Copper, Bronze... maybe, I don't know, probably fine.
Gold and Bendalloy and likely Cadmium too are prohibited for filling because ‘Oneg Shabbat. Tapping might be allowed. Electrum and the Spiritual metals are also probably fine.
You are right on the money with hemalurgy not being allowed on Shabbat. Even if it didn't involve murder, shedding blood on Shabbat is a problem - which, funnily enough, is coded in the Simple Rules on Threnody. Fun stuff. Anyway, no stage of the spiking is allowed, but I think most uses afterwards are rather passive so no need to consider that.
Sel magics are forbidden, but not for the reason you think. It's prohibited to write on Shabbat. Both forging and the Aon Dor require writing, even if it's sort of temporary and in the air. It's also not always two letters, but the writing systems used aren't alphabetical so it might work differently. Remind me to look up how writing Chinese is treated by Halacha. Though if we're being honest, the distinction between one and two letters isn't relevant in the modern day.
Awakening is creating life. G-d has rested on Shabbat from creation. That, admittedly, isn't too well-based on the halachic reasoning, but there's precedent to Golem creation and usage being disallowed on Shabbat so I'm going to stand on a firm no.
Personally, I think on some of the things I voiced an opinion here there could be deeper reasoning either way. It's actually all very complicated. So I think some Invested arts would be allowed and some not, but I don't think it's as straightforward as you listed it. However, it was a fun question to discuss! There's more to touch there too.
In addition to that, things like surgebinding might be a problem because of oaths. Judaism isn't big on swearing oaths in G-d's name too much, and is big on not swearing to foreign gods. So the Radiant oaths might be a problem. Now, of course, with how Roshar treats oaths swearing them to G-d might be fine. However, I'm not sure I won't to know how this clashes with Hatarat Nedarim.
On another note, anonymous asker, if you're not someone I know who's already involved in it, maybe you'd like helping with working on the Inexplicable Cosmere Judaism in general? Maybe if we have enough people involved I'll start a new discord server for that, actually. Right now it's mostly me and @springstarfangirl, and we're working on a thread in a shared server as well as in DMs and other random places, but I'm open to other people making contributions!
