Your website indicates that it isn't such a great idea to remove the movement and immerse it in a sonic jewelry cleaner. But what would be the harm in trying that before incurring the expense of having it cleaned by a watch maker? I would have to take it to a jewelry store, then have them send it off to a repair person. Just the turn-around time is at least two months.
There are a few problems... One is that without disassembly, the cleaner will not get all the pivots clean, inside. It will leave old oil and debris behind to a certain extent. Secondly, you can't properly replace the lubrication without disassembly. Just like a car, oil must be everywhere it should be and no place it shouldn't be, otherwise it will run "fine", right up until the pivots grid away enough that it doesn't. In addition, it is very unlikely that all the cleaning solution can be removed afterward. Especially inside various parts like the mainspring barrel, leading to rust. Lastly, an ultrasonic cleaner will likely get in around the shellac that holds the roller jewel and other parts, knocking them free. The pallet fork, balance, and hairspring are properly cleaned separately using other techniques, as is the mainspring.
Believe me, it's not a good idea...