I would think this is a garden forum question.

If the lilac is small then yes, your dog peeing on it all the time could damage it. Have you been watering it regularly?
Lilac's are prone to fungal and bacterial disease so I'd take a cutting into your local nursery to see if it can be saved.

I can pass along this information from having made a mistake with two lilac bushes. They do not like to live in very wet soil. I killed one lilac, but managed to rescue the other by transplanting it from the wet area to a well drained hillside.

Another thing is that lilacs like alkaline soil, so if you have acidic soil as I do, it is a good idea to put ground limestone or wood ashes around them periodically.

I'm sorry for the loss of both kitties, now and in the past. I think planting a flowering bush on the spot(s) is a lovely memorial.