That inverter will draw 36A from the 12V battery plus more to cover whatever is lost (36 + Ploss / 12)A.
I would use a full-bridge, possibly with a few parallel MOSFETs per switch, to step that up to about 350V, and then a PWM inverter to get the 220VAC out of it. The step-up will obviously be the hard part, as the current will be fairly high; the inverter section, at just 2 amps, won't have to be very special at all--probably just a pair of IGBTs wired to a basic PWM (it will have to have a little dead-time though). Gate drive on the high side might be a little tricky because of the widely variable duty cycle (you can't use a simple pulse transformer in this case). Carrier drive, or a small floating supply, would work.
If I might make a suggestion, though, does the battery side *have* to be 12V? The UPS systems I've seen used 4 smaller 12V batteries instead of one big one, to give 48VDC. If you do this, then your low-voltage side current is only 9 amps, which is much easier to switch.