there are 3 plausible ways that our protagonist can increase the power + longevity of the heat in my personal opinion
1. Force
this one is pretty simple, just increase the power/speed (like making the muscles responsible for shutting down stronger) at which the claws close more powerfully, to increase the heat and the lifespan of sun level heat by a few seconds.
2. Material
like force but instead change the material of the claws to metal to create a point of intense power. it can vaporize air and destroy whatever it slammed down on. (when combined with the force method it can create enough power to puncture through Cole with ease and depending on the size of the claw, it could completely cleave Cole in half)
3. Pressurization
scraping the singluar point of contact and focusing more on RogueEagle's idea we can instead the replace the hammer and anvil single fire system with a pressurized chamber inside the Protagonist body which can compress, retain, and release heat through a hole (He would need to find to compress air but more importantly keep contained airtight [maybe make the chamber heat-resistant/heat-proof as to not damage him]) this would be more in line with "the solar bean" that RogueEagle is talking about.