Co-generate with a Stirling engine or steam turbine. Your application is the wrong scope for TECs. It will be a hundred times more expensive, and a tenth of the efficiency.
DC doesn’t like long runs, but I believe there’s a battery in that yellow block along with the charge circuitry. If I’m wrong it may warrant another look.
https://www.bioliteenergy.com/products/campstove-2
The idea of putting my phone, or any lithium battery really, that close to fire... this plan is not for me.