Which MOSFET is best for me?
Head over to your favourite parts supplier and enter "trench fet" into their search engine.  I'm partial to Digikey but many other suppliers have similar search engines.  

I think that you will find an awful lot of suitable MOSFETs to use. Choose one that has a threshold voltage down near 1V and then look at the datasheet to ensure that it is fully enhanced with your control voltage HI level.

FWIW - trench FETs are a class of MOSFET that has very low threshold voltage, very low Rds on, quite low max Vds voltage.  Some are available in tiny packages with awesome current capability.

You should not need a heatsink if you choose the right FET.