Monzo are going to build a virtual savings pots feature which will be their version of savings accounts, they’re on the roadmap here.
So if the app was built to enable this, I assume that you could select the saving pot that you want your change to be moved to & then use the funds from that pot, in whichever way you choose.
I’m not sure what you’re anticipating there though - there shouldn’t be any need for a provider to add extra charges in order to integrate with Monzo.
So I would expect to pay the same fees that I would pay currently, if my change was being invested with someone like Moneybox or none if it was being donated to charity & the app / integration was set up by a selfless developer.