I live next to Avondale, PA (mushroom capital of the world), and we looked into having our manure hauled away by mushroom growers.
First of all, most growers will not accept manure mixed with wood shavings (it takes too long to decompose), as they prefer straw. For this reason, we bought a manure spreader.
Secondly, I have never seen a grower show up with a tri-axle dump truck, at least not when hauling residential/small farm manure. Most locals use the equivalent of an F-350 or Dodge 3500 dump truck (but I'd check with your grower).
Thirdly, as cowboydoc pointed out with his 4700, I have a JD4200 and I'd have no problem either.
I'm betting you'd be perfectly fine with the tractors you have in mind.
Good luck, hess.