Seems straightforward to me. McGonagall knows that she does not recall being handed 100 Galleons an hour ago, so the three states of the world with high probability are: 1) She is not in a universe where she will hand Harry 1 Galleon, 2) She is in a universe where she hands Harry 1 Galleon and Harry breaks the agreement, or 3) She is in a universe where she hands Harry 1 Galleon and Harry keeps the agreement in a way that leaves her unable to recall this happening. By not handing Harry a Galleon, she will ensure that she is in universe 1. By handing Harry a Galleon, she will find herself in universe 2 or 3. She should therefore give Harry a Galleon if she judges it less than 99 times more likely that Harry will break the agreement than fulfil it in a way consistent with her experience.

As Harry has access to a time machine, he doesn't need to decide to give her 100 Galleons before he gets the 1 Galleon, so the situation is quite different to one based on predicting her actions, as Omega does in the Counterfactual Mugging. Rather it has most of the properties of the forward-time version of the gambit: "If you give me 1 Galleon now, I'll hand you 100 Galleons in one hour", except that McGonagall has a big piece of evidence that the promise will be broken, namely that she doesn't remember it being kept.