Drilling an extra hole in the leg will weaken it to some extent but the significance of that depends on the size of the holes relative to the diameter of the leg, how close the holes are together, the wall thickness of the leg and the loading it is subjected to in use.

In normal use it's unlikely that adding one extra hole would cause a problem (as long as the holes weren't very close together) but in this day and age products are designed to a price and the minimum amount of material is used to save on cost - so the tubes making up the legs are probably quite thin walled. Tubes are naturally strong structures resisting forces evenly in all directions - cutting holes into the walls weakens them.

However, if you were to slip whilst in the action of sitting down and landed heavily on the chair the the shock loading could cause the leg to buckle or fracture and the weakened point - you'd hope that the design would accommodate that sort of scenario but ......

Wet rooms have gently sloping floors so I can see how your problem arises - the adjustment for the height of the legs is too coarse for the slope of the floor. Would it be possible to turn the chair so that the floor slopes say from back to front underneath it rather from side to side? Probably not but it's a thought.

To cope with the gentle slope of a wet room floor the feet of the chair would be better fitted with a threaded shank which allows the foot to be screwed up or down to exactly the right position. Presumably your chair doesn't have this type of adjustment in addition to the pin and crutch method you describe?

Failing that, sticking on the right sized shim to the underside of the foot sounds good to me - I work in engineering and much as I love technology I guess I'm "old school" - the simple solutions are often the best!

Steve