How do fees get processed in bitcoin? (protocol question) If the fee isn’t written into the ledger, and it’s just the difference between the input transaction amount and the output transaction amount, how does the miner prove he has access in his wallet to the fee amount?
how does the miner prove he has access in his wallet to the fee amount?


Miners do not prove this.
The ability to reap the difference in inputs and outputs in a coinbase transaction is baked into the bitcoin consensus rules.
Any miner who was lucky enough to find a block can sum all the differences in included transaction inputs and outputs in a block and aggregate them in the coinbase transaction.

The miner can then spend the coinbase outputs 100 blocks later.

Edit #1: @eponymous's comments regarding potential reduction in bitcoin supplies if miner does not reap all transaction fees.

Suppose a miner elects to not reap all the difference in inputs and outputs in a coinbase transaction, then the supposedly total supply of bitcoins will be reduced. The 21M bitcoin supply is a theoretical limit that assumes that all miners reap all potential transaction fees and block subsidy rewards. 

Example: Total difference in inputs and outputs is 1 BTC, but miner reaps 0.9BTC. Then 0.1BTC will be lost from circulation.

Edit #2: Included @pwuille's excellent explanation on the existence of 21M bitcoins