Go around short circuit protection
The battery spec says:


  What sets this battery apart from our other 110mAh battery is that fact that it was truly made for e-textiles projects. With no risk to short out and a low current delivery, the E-Textiles Battery will cut off in the range of 240-380mA making this the safest option to power electronics sewn into pieces of fabric.


The cooler spec says:


  lmax - 7 A.


That battery is not capable of powering that cooler.