I wish I knew more about HST transmissions because it sure sounds to me like that is where the problem is. There is a high pressure pump in there, and it is a major component.
Lots of us old mechanics know standard gear transmissions quite well, but not much about HST even though they have become very popular. In spite of their complexity and requirement for nothing but the purest trans/hydraulic fluid, HSTs have been very reliable. Argueably more so than a clutch and manual setup.
But it sounds like yours either needs a replacement cable or needs an internal rebuild. Here is what happens normally: As you push down on the HST pedal you are allowing a "wobble plate" to tilt which in effect changes (raises) the gearing by allowing a high pressure pump to send more fluid to the drive motor. If that wobble plate moves - but the system doesn't have have the high pressure fluid available - then it is as though you shifted to a higher gear and let off on the power at the same time.
But it might not be the high pressure pump at all. It could just be the cable that moves the wobble plate. That cable goes from the HST pedal to where it wraps around a drum that turns a shaft that moves the wobble plate. I broke that cable on mine - which is how I know about it. A one hour fix. You can see the cable and see if it is doing its job by crawling under the tractor.
On most HST trannys there are ports on the transmission housing for measuring HST pressure. If it isn't the cable, then making pressure measurements is where I would start if it were mine, I would get a real good OEM shop manual for that model, make up some pressure guages (cheap to do) and make some HST pressure measurements. Of course a dealership mechanic might be able to do that too, but chances are they've never rebuilt a HST. Today they just replace them. And it may come to that as well.
Good luck,
rScotty