Well here is a suggestion.
Have the Fabricator remove the Backsplash and have them cut down the back splash 1/2" to remove the stain and the install the Backsplash with a 1/4" spacer at the back
then you would not have the problem in the washroom.
For sure the installers used the wrong type of calking . Maybe exterior caulk or roofing caulk? Ive been Installing counter for 10 years and never has that happened to me even if I do the silicone before we use impregnator /
Sealer.
You can try to dry it out with a heavy duty Heat gun before and against the wood Gable in the Kitchen.
adjusting the Backsplash might e the easiest option.
Let me know how you end up.