> Transaction fees are based on the complexity of the inputs/outputs, not the value transacted

Not on the lightning network. Fees are used to incentivize or disincentivize routes across channels.

> The institution handling this offchaing lightning branch can implement fees in whatever structure you agree to transact, including percentage based.

No institution is needed. Even if one is used as an intermediary, when using lightning non-custodially, the economics of lightning are such that fees are determined by the nodes in the payer's desired route. If it's a custodial transfer from one user to another, no routes are needed.