How to choose/find the right H-bridge IC

Hi,
I am working on improving an already existing system which keeps the temperature of a photodiode constant by measuring its temperature with an RTD temperature sensor, and heating or cooling it with a Peltier element via an H-bridge. The control algorithm was done with the Arudino PID library.

One of the changes I want to make to the system is to replace the H-bridge (made up of 4 MOSFETs) with an IC to perform the same function. The power supply I am using is at 5V (max 2A), and the output current that the H-bridge supplies to the Peltier has a maximum of 1.6A, but I may change this to 3.9A if possible (based on power supply limitations). I am finding it hard to find H-bridge ICs that are suitable, as they seem to either operate at a much higher voltage or a lower current.

The MOSFETs currently being used are: http://www.mouser.com/ds/2/149/FQP30N06L-112819.pdf

Could anyone please advise me where to look for this kind of thing? The closer to the behaviour of the IC is to that of the current system, the better.
Thank you very much

This simply isn't going to work, discrete MOSFETs have way better performance than MOSFETs integrated onto ICs.

Discrete MOSFETs use vertical current flow through the chip for much lower on resistance (100's of times
lower is possible), and this cannot be integrated into a single chip as the substrate of a chip is electrically
active over the entire chip. So MOSFETs on an IC have to be horizontal current flow, typically down to 0.2
or 0.15 ohms minimum, whereas discrete MOSFETs can be 0.002 ohms or lower...

There are some clever packages which combine 4 MOSFETs and control IC into one package:
VNH5019 for instance. This might be what you are looking for.