Yes, go ahead and do it. An ultrasonic batch is great for all parts, including shifters - STIs, Ergos, the lot. All you will have to do afterwards is remove the idler wheels and dry the axles/bushings and lube them inside there. I have no idea what Jizer is but any soapy degreaser will do, no need to use a petroleum product - dangerous in an ultrasonic bath in anyway.
As for cleaning shifters in such a bath, Shimano products are fine but Campag Record ergo levers have a ball bearing inside that requires special attention once it's been in such a bath. I just disassemble it and rebuild from scratch, using the ultrasonic bath to clean the individual parts.
The ultrasonic bath works perfectly for chains as well, I use it primarily for that.