Something to consider (if you're haven't already): A Charmed/Shadowhunters crossover. They both kill demons. It could reasonably work out that they both meet at some point. And given the fact that Grams couldn't keep a man because they were generally horrible in one way or another, I could easily see her having dated a Shadowhunter at some point. Also, Isabella meeting meeting the three most powerful witches on the side of light who also happen to be some of the most badass women around??? Definitely feel like they could get along, if not be friends.
This is tricky because those are both franchises with established lore on how magic users work - and that lore is inherently contradictory with no room for compromise.
There's the more minor things, like how warlock is the main term for witches on Shadowhunters but the word warlock had more of a negative connotation on Charmed. The way they use magic - Charmed with spells, Shadowhunters with just their natural magic. The existence of warlock marks. Mortality vs immortality.
And then the big stuff. Like, where witches and warlocks come from. On Charmed, the magical lineage is very much an important part of the show and they serve the light and are naturally linked to heaven-like beings. On Shadowhunters, warlocks are distinctively the offsprings of demons - and are sterile themselves, making a lineage of witches impossible.
To make this work, either the warlocks on Shadowhunters would have to give up their warlock mark, immortality and the burden of the demon blood in their veins as well as the societal implications that had always been linked to that demon blood (-> the racism from the magic angel squad cops), to fit the Charmed lore on magic-users, or the Charmed Ones would have to be the children of a Higher Demon and the long-lasting lineage of the Warren family would have to be erased.
Which, granted, on account of being a Phole shipper I am going 👀 at the idea of adjusting the Charmed universe to the rules of Shadowhunters, because Cole is already canonically half-demon, which would make him thusly a warlock and his Belthazor form could easily be his warlock mark, and all the bullshit about why Phole couldn't work out would just be out the window because Phoebe, Prue and Piper would also all be half-demons on account of being witches.
Okay you know what, I think I just sold myself on this after all for shipping reasons. Damn, I'm weak. When I started this reply I was firmly "no. they don't work in a way that mashes, one lore would have to be erased" and just one thought of "but my ship may work out this way", and baaahm I am on board again.
Though would that make White Lighters, as the ones with the connection to heaven, Shadowhunters in such a translation, seeing as the heavenly link for witches would be gone? Making Piper and Leo a romance between a Shadowhunter and a warlock?
(I'm unsure how Paige would fit into this all to be honest.)