If OP needs to have them serviced to feel comfortable, then by all means. But you can inspect and and function test your regs very easily. After washing, put them on a cylinder, leaving the valve closed. Or, securely tighten dust cover. Either way, just so no air comes in through the first stage. Try to inhale on a second stage. You shouldn't be able to -- that means the exhaust valve is working. If you get air, then you have an issue. If you have second stages with seat saver (Atomic/Zeagle F8), you might get a little air coming into the system from the other second stage, but otherwise, it should be easy to tell that the exhaust valve is working. You can check the purge diaphragm by unscrewing the cover and looking. Or, look at the service manual for your second stages and you can see how to access the exhaust valve and look at it yourself.
But, like I said, if the regs need service anyway, or it makes you feel better, go for it.
A basic function test is easy. Does your first stage hold IP? Does the system hold pressure after you turn off the tank valve? Do the second stages free flow? Can you pass the exhaust valve test above? Do they breathe okay without giant drops in IP? The whole mess takes about 15 minutes. If your regs check those boxes, you're good to go.
Just be aware that, either way, you will need to do the cleaning part yourself. "Service" does not include sterilization. At most, the plastic parts get a quick rinse with some dish soapy water. Only certain of the metal bits get ultrasonic, and only if they need it. The exhaust valve is generally only check-and-replace if needed. It generally isn't in the parts kit for regular service. Same for the purge diaphragm. The parts most likely to be impacted by a curious mouse are not those that get replaced, are easy to check yourself, and you're going to have to clean them yourself anyway because the service tech is not typically do much to them.