I have 2004
BX23 (I bought new)with 802 hours. When I start out from a dead stop and push the foot petal hard to go forward the tractor has a very noticable viberation or shimmie in both high and low.
Shimmy: shaking, esp. abnormal vibration of the wheels of a motor vehicle
A true shimmy takes some speed to occur and you state that yours does this as it begins to move. I'll take "vibration" as a better description although I understand how difficult it is to define a particular problem. One thing I can bet you though is that if the tractor has a noticeable vibration when you move, something is not right.
My observations:
The dealer had it for four weeks and couldn't isolate the problem could translate that he had it sitting around for three weeks, four days and six hours before he looked at it and couldn't come up with a quick and serviceable answer. I'm not saying that's the fact but the length of time he had it means nothing.
"When I start out from a dead stop and push the foot pedal hard to go......" That seems to be an unusual way of operating a hydro transmission. The statement caused me to think how I operate mine and possibly other than repetitive loader work, in low range at a reasonable RPM, I don't start off depressing the treadle all the way. Could doing so have created a problem? Darned if I know but the control pedal is not, as you are aware, similar to a gas pedal. It controls torque and speed, less of one, more of the other.
Were it my machine and the vibration was sufficient to get my attention, I would look for a problem with running gear, U-Joints, final drive gears, brakes, lug nuts, clutch, something, anything downstream of the actual pump and transmission.
Finally, back to the dealer: If I had a "bad" vibration and a Kubota dealer couldn't figure it out, I'd probably be looking for a next Kubota dealer. Usually living with something like this instead of fixing it only leads to a bigger bill, down the road. In your case, if it's annoying but not severe, I'd consider living with it to see if it gets worse which may make a diagnosis easier.
Here's hoping you find it and it doesn't cost you a bundle to correct.