There's an old thread about someone doing it here: https://diysolarforum.com/threads/replace-apc-ups-sla-batteries-with-lithium.25424/ but sadly he doesn't give any details, or any updates.
It's wierd. APC do a really crappy small one for telecoms cabinets, but none for servers. I would love to replace my APC batteries with lithium ones.
Let us know what ones you chose in the end.
I don't think so. Cheaper batteries have that problem, but a decent brand does not. Check out this one: https://www.powertechsystems.eu/home/products/48v-lithium-ion-battery-pack/48v-105ah-5-38kwh-lithium-ion-battery-pack-powerbrick/ I bought one for my house, and have a 5KW inverter connected to it. Its specifications say that can do 120A drain continuously. I have used it to boil my 3KW kettle a few times in one day (but not often - I usually use the power for other things), and it has been fine.
That's just a limitation of the product, not the technology.