Based on the fact that you have to pull it out of reverse then re-engage reverse, I am thinking something inside the gearbox is amiss. a gear is probably floating on a shaft and is not fully engaging, then pops out, pulling it out of gear may allow it to shift back closer to it's correct position, re-engaging gives another bite on it. We had a similar problem on one tractor that failed our post assembly testing. That one would not stay engaged, as soon as you stepped on the clutch it would release, we would pull it out and put it back in gear (cant remember if it was forward or reverse now) and it would go until the pressure on the gears was relieved when the clutch was pushed in. A similar problem in your tractor might explain why it happens when backing down a hill, once gravity overtakes the tractors own drive, the pressure on the gear train is removed and things can shift. In the one we had, it was a snap ring out of place allowing an idler gear to slide off its bearing. Of course it is deep inside the front portion of the gearbox (shuttle area). You coule try driving on flat ground in reverse stop and step on the clutch and then restart a few times and see if you can get it to pop out under those condidions.