Actually the way to settle differences between the branches is to use the third branch, the Judicial branch. Congress takes them to supreme/federal court, which decides if the subpoenas are valid and would then enforce the subpoenas and the executive branch then has to comply. Basic checks and balances.
https://www.theatlantic.com/ideas/archive/2019/12/house-demo... https://www.theatlantic.com/ideas/archive/2020/01/democrats-...