I think hydrogen it a pretty poor coolant anyway. They could presumably use a mixed fuel or use an inert gas like argon for cooling.

If they wanted to be "fuel rich" they could just release some excess oxygen as process waste.

So that's not why this is a dumb idea

The reason this is a dumb idea is because eloctrolysis is the hardest and least efficient way to get hydrogen.

And also because hydrogen is a really hard gas to store and handle, which is why every hydrogen effort going back to the Bush administration has failed.