How to run two Postdoctoral fellowships at the same time?
Generally foundations "request" that you inform them if you receive extra funding for the same budget. They probably expect that you apply to several foundations simultaneously. However, if you negotiate with foundation I to change the scope of the funding (such that it doesn't go towards your salary, which is presumably already covered by foundation II), then I think you're in the clear.

Alternatively, if foundation II didn't fund your full budget, then you could accept part of the funds from foundation I to cover the remaining costs, then inform both foundations of the arrangement. That should also be uncontroversial.