Certainly all sounds like an issue with the MOSFET specs, I have done loads of testing with FET's and 3.7V builds but unfortunately this isn't a FET I have tried.
I have had the same issues running a 3711 FET under high load with round cells due to high volt drop. If the Gate-Source voltage is not sufficient (more than the Vgs MAX of the FET) the FET will get hot and full current will not pass to the 510. The characteristics of different FET's varies quite dramatically, I am no expert but have spent a lot of time playing with and testing FET builds.
From what I have played with-
I would recommend the IRLB8748 FET for use with standard round cell 3.7V parallel builds. I have tested plenty and these are my preferred one for standard round cells at 3.7V. These still get warm running 60-70W+
The IRF3711pbf is great for LiPo builds as it starts to cut off around 3.0V under load which helps protect the batteries from over discharge.
I am sure there are better ones out there (please share if you find them!), but I know these work and have a good idea how they will react.
Remember- 2 FET's are better than 1, Heat Sinks are cheaper than chips and are very helpful.
Don't forget FET's will generate heat and you need to allow that out so wrapping in insulation is not ideal for dealing with high loads/more heat.
Hope that helps