I have been dealing with landmen for 6 years, most of them leasing property in the Permian basin. I don't think anyone has ever offered a 3/16 royalty to me in that time. I do have some at 3/16, but they were in the 1980's. As I understand your situation, you have an older lease holding the land by production and they are at 3/16 leases. If you were going to sell the property, you might want the buyer to recompute a price at a 25% royalty rate, since the existing leases will eventually be gone and new ones which a knowledgeable executory rights holder would negotiate at that higher rate. just a thought.
Good luck.