Update from me:
I ordered a Portable Power Station that has UPS functionalities. There seem to be a lot that do pass-through charging (Jackery, Yeti, Goalzero etc.) but that basically continuously degrades the battery. Ecoflow power stations (like this one) bypass the battery when it is fully charged and directly provides power from the outlet to whatever device is attached.
On the topic of why a UPS doesn't really do that job: The type of battery most UPSs use (lead acid or gel) are meant for high peak power rather than slow, deep discharge. That's why even 900w UPSs oftentimes only provide power for 3-4h for low wattage appliances.
Since this is a lithium battery, it shouldn't have that issue. I'll post an update on how long it can run my setup once I have it.
Yeah definitely keep us posted! I'd be curious to see how long it will run your setup!