I've never seen a rat with head tilt, but I have, unfortunately, seen them with medically-unresponsive illnesses (which may be contagious to humans).
I'd euthanize. Some years before I ever got into keeping snakes, I had a few pet rats (-when I was in college, easy pets & no yard needed). My initial
ones were healthy & remained so, but later (then rat-less), I bought 2 females from a different pet store. Both came down with a respiratory thing, I
took them to the (very surprised!) vet, and when both courses of different antibiotic treatments failed and they died, I want to say he looked concerned,
but it was more like fear. He actually suggested I not get any more pet rats.
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