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If you're vegan it's no go as hide glue (even the modern stuff) like almost all traditional glue is made from boiled bits of dead animal I'm afraid.
Also it's not really appropriate for what you would be using it for as condensation would eventually cause the glue to fail.
I'd out a thin bead of glazing putty either traditional linseed oil ones or modern mastic tube stuff. Behind the bead and then fix the beads with panel pins.
Not late as I've not actually done anything yet!
I'm vegan though so hide glue doesn't sit comfortably with me (not that I know for sure modern variants are made with animal substrates).
For context this is for refitting mouldings around glazing in an external door. They were originally held in with what felt like a type double sided tape. It was strong enough, but pulled apart relatively easily and was somewhat elastic. I think I'm just going to use veneer pins when I fit them back.