Jacob, nice idea but too complex.

Easier to use a LH2-based evaporative cooling system, since you've got so much of the stuff anyway. Then you don't need seprate pipes and tanks for LHe.

If you're going to the trouble of adding extra piping, better to go the route of a proper refrigerant cycle with LH2 as the heat transport medium.

I admit that there can be some surprises in the refrigerant world - I was badly wrong a while back with an IR telescope that used SOLID hydrogen as the cooling material - cooled so far down that it wasn't a liquid anymore. I assumed the author of a press release was talking through their hat but they were correct. The telescope was operating at very low temperatures and using the solid-liquid phase transition as an additional latent heat store (plus some extra sensible heat as well)

That's one way to extend the lifetime of an LH2 system, although you'd want to be sure it was all liquid before you started burning it for fuel and expecting it to flow through pipes and turbines.