Cross-chain aware contract without mint function?

Hello, how do you create a contract able to manage part of the totalSupply in a different blockchain?

I know a bridge can mint/burn supply in both sides to keep the overall amount constant, but this would require a mint function in the token, which is not well seen by investors.

Is there any alternative approach to bridges to maintain the totalSupply between 2 tokens?

How do you maintain part of the totalSupply in BSC network and part in ETH network?