Probably better off finding an actual UPS for cheap. I got an old Tripp Lite rack mount (with expansion) for $100 with no batteries. Paid about $300 for new batteries and now my 300W rack can go over 10 hours without power.

Your DIY method seems needlessly complicated, but I do have something similar for my sump pump backup. 12V AGM battery hooked up to a transfer switch with an inverter. I just use a noco battery charger, but really any trickle charger will work.