The suitable OPs depend on the supply and the use in the circuit. The OPA277, OP07 and OP27 all need a negative supply.
The buffer for the cheap DAC is not that critical - here it's more the question if one needs one at all. So the OP0 would be well good enough here.
The critical OP usually is the one amplifying the low voltage signal from the shunt. This could be the same OP driving the MOSFET, but could also be an amplifier stage first.
The OPA277 normally is rather expensive, usually more than the OP27 and way more than an OP07.
The OP27 is rather fast and could thus tend a little more towards oscillation.
With no scope, it would be a good idea to first simulate the circuit. One might also use a simple AC detector to at least get an idea of the amplitude of possible AC signals.