Need help to find specialized watch repair

[Cat City Slim]

I need to find a jeweler or repair shop to change out the battery on a Hyperbar System dive watch. As you can see in the photo, the cases of these watches are filled with liquid. This renders the watches waterproof, as the liquid pressure inside the case will equalize with water pressure. The watch manufacturer seems to be defunct. Can anyone suggest a repair shop that might have experience with this type of watch? If the liquid is spilled or contaminated it will probably destroy the watch, as I don’t know if I can get it replaced. Someone in the Tulsa area is preferable, but I’m willing to send the watch out if needed.