This glass mug by Bodum has a rubber stopper at the bottom that is designed to allow the air trapped between the two layers to expand due to heat but to not let water in since the water would get trapped inside. Can it be toveled/is the rubber a hatzitza? It is designed to be there and be left in place, so would it be considered to be eino makpid? (YD 121:13)
If this is a functional part of the mug, then it is more than eino makpid – it is part of the mug itself. This should be no different than, for example, a pot with a wood handle – do I have to remove the handle so the water can get on the surface area of the pot that is being covered by the part of the handle that is attached to the pot? Certainly not.