clean copy of ios 16 but not lose my data and apps

my iphone 13 pro is overheating and started around the update to 16.4. i only just got a pop up warning of to hot yesterday. Apple store refused to replace phone saying doesnt appear to be hardward issue. it appear software. like i didnt already suspect that. they only want to restore my phone manually thinking i have a corrupted software update. however, years ago on another phone i went through that and still phone overheated.

If i do a full back up to my macbook and then do a factory reset to my phone and once up i then restore from my backup on my macbook will that fix the software? Making it a clean copy of my ios system to see if it will stop phone from overheating? Or Will i have to get my cell carrier to really get involved and honor my added warranty on the phone and replace it for safety reasons..