Well at least they made a trade! Most of the time they simply act like thieves!
Maybe you should train the animal to go on a gluten free diet!
Or you could make a hardware 'cloth' (half inch 'weave') cover for the gauge?
BTW, does your rain gauge have the anti-bird spikes?
We had a similar problem with raccoons many years ago. We have some powered attic/roof vents that have some poly-screen material to keep out bugs. Didn't bother the raccoons at all, the screening is very flexible and just got pushed aside. I ended up making hardware cloth covers on the inside, under the opening in the roof. I wasn't sure they raccoons had "left the building"! We never smelled any indication that they were still inside. I can only assume they were out visiting friends when I 'locked' their entrance.