I’d put the speaker in the sun for a few hours and out in the fresh air on a low humidity day. You could cover the wood with a blanket if you want to make sure the finish isn't somehow changed by the sun. If that doesn’t do the trick to rid it of the smell, then as others have noted, it may be time to replace the driver and call it a day. Be sure to "break in" the new driver for a few hours to get it to sound matched to your other one.
Then again, if you are going to all the trouble, maybe take a look at the crossover components used inside BOTH speakers and contemplate an upgrade if it makes sense. At least you are lucky and should be able to get a replacement driver from Klipsch since these can’t be very old.
As I’ve noted before on this forum, a friend of mine had kittens that were doing a number to the grilles on his then new ESS Heil Air Transformer speakers.
His "fix" was to make some nice frames from chicken wire, spray painted them black, and affixed them to nicely stained wooden frames at the bottom that matched the finish closely of his speakers. They worked well and didn’t seem to affect sound quality at all. In fact, they looked pretty cool and kept any small visitor children and curious, but clueless adults from touching his speakers.
Good luck. Dogs are cool and can be taught. Perhaps an obedience class is in order to make your (and the dog’s) life better in the long run.