Not an expert either but I should keep it somewhere well ventilated, where the glass isn't going to get broken and avoid sitting too close to it for extended periods. If the radiation level is close to the normal background at a couple of feet, that's probably a sensible distance to keep!
I have a pocket watch a lot hotter that lived in a pocket for decades with no apparent ill-effects to the owner. Provided you don't get radium paint dust on or in your person (or house), or breathe in a noticeable amount of radon, you'll be fine.
A worthwhile test might be to put the meter in a ziploc bag for a couple of hours, then removing it to measure the half-life of whatever gas is left in the bag. There might not be anything detectable on your GC but things like altimeters and ASIs are open to the atmosphere and leak quite a bit.
If you're not seeing a difference when you put a sheet of aluminium in between the meter and the GM tube, there's probably a lot of beta radiation your meter can't detect. All the alpha particles, which stimulate and ultimately destroy the zinc sulphide phosphor will be comfortably contained behind the glass, but any leaking radon will also emit alpha particles so you don't want to be breathing the stuff unnecessarily. Polonium is a further decay product you don't want to be near.
https://www.ld-didactic.de/software/...a226Series.htm
Colin