How to effectively control a Peltier device with MOSFETs? (general design and heat issue)
Your problem is that you are using all N-channel MOSFETs. The high side ones are being driven by +12 so they act as source followers and drop several volts.
Q2 and Q4 are the ones in question.
In order to avoid that, either drive them from higher voltage than 12V and add circuitry to make sure the +/-20V Vgs(max) is never exceeded, or use P-channel MOSFETs for the high side and ensure shoot-through is avoided.
They’re not especially low Rds(on) MOSFETs, sub-milliohm ones are available and those are 500x worse.

Edit:
I mentioned other MOSFETs (do a parametric search to find them, at any distributor such as Digikey) sub-milliohm Rds(on) above, and your one is ~550m\$\Omega\$ (when cold, much more when hot) so that's &gt;500x worse than 0.001\$\Omega\$.
You won't even notice it until you fix the source follower thing, but 2 MOSFETs running at 1.5A and 0.550 ohms and maybe 40% more because they are hot will dissipate a total of more than 3W. A pair of 0.005 ohm MOSFETs (100+ times better) will dissipate a total of 15mW, which is negligible.