There are quite a few different chip-type amps, boards anyway. Most of the assembled units seem to have level controls - although I suspect they could be easily bypassed.
What I'd do is find a few likely candidates then find the chip manufacturer's datasheet. Those data sheets normally have graphs in them showing output power vs. supply voltage.
Personally, I'd think there wouldn't be much difference between them in terms of power at a given load impedance for a given supply voltage. You might see a meaningful difference if you look at PBTL vs BTL configurations but you only get the higher potential power of PBTL if your speakers are low impedance. The graphs in the datasheets will show you that.
The power supply you mention shows 12V as the output so that's what you need to go by when looking at the datasheets. Then, once you find the power point you have to then consider if the 2A current rating of the power supply can actually support that power level.
For one example, the TPA3116 (using the 1% THD figures) -
With 12V power supply you can get roughly 8wpc @ 8 ohms or roughly 15wpc @ 4 ohms.
Your power supply current rating of 2A could support the 8 ohm rating as that would require ~2A current for rated output. However, while the chip is capable of 15wpc @ 4 ohms using 12V supply, the 12V supply you have would struggle with that rated output because it needs ~4 amps from the power supply.
