Boost converter with battery issues
Some thoughts:

What if you tried cooling the battery pack with a fan (and avoid cooling the evaluation board and/or load)? If after the "7 minutes" the input current is lower than "almost 4 A" it is almost certain due to the temp of the battery.

If the internal resitance is indeed 1.65Ohm, note this is also very close to the UVLO of 4.25V (11-4*1.65 = 4.4V)

I read something in http://www.ti.com/lit/ug/snva461a/snva461a.pdf
at section 4.5. Maybe that is applicable to this situation?