You don't divide the subsidy. You only divide the reward, part of which is the subsidy. The real question is: What will happen if someone wants to move an amount, and pay 1 sat as a fee, given that 1 sat is extremely valuable.

Even though fractional satoshis could likely be implemented in a soft-fork way, it'd be preferable to do it in a hard-fork way. And since ECDLP will be likely broken by 2136, a quantum-resistant algorithm is going to be needed, which will result in an inevitable hard fork. Perhaps we could expand the sub-units along with that.

This problem will be a serious concern only if there's a tremendous rise in production of goods, because otherwise, for a hyperinflated USD with little rise in production, $10 isn't going to be much (which is the equivalent of 1 sat, given that 1 BTC = $1B).