I don't think that there's much point in having two inductors. The overall inductance and the saturation currents are the important considerations. Similarly, there's not a lot point to having the capacitors in parallel with the Peltier cell, as the current is kept reasonably constant by the inductance. You need lots of capacitance on the supply to the H-bridge, when using PWM, as the input current is discontinuous.
The important consideration is to make sure that the H-bridge is driven in a way that shorts the output during the off part of the PWM cycle. On the MAX14870, that is done by using the PWM input, and not a direction input.
The MAX14870 has a maximum current of 2.5 A so it won't drive 3 A into the Peltier cell. You need a larger H bridge, but you might be able to parallel two bridges.