As much as I would hate to admit. Yes, bore chemicals will damage the bore scope. I thought that they would be sealed, but they are not. Chemicals will, can, and do get inside the camera and light unit and damage it.
Lyman has awesome customer service though and did replace mine... twice. Please dry patch your bore as you check your progress.
I spray the q tip with the cleaner before cleaning the mirror. I don't apply the spray direct to the unit.
Even if I have lightly oiled the bore, I will dry patch it once before using the scope to check it. It seems a bit anal, but better safe than sorry.