Torque loading for the axles is a whole other proposition from the load carrying aspects. If the tires spin, generally speaking, you haven't got enough weight on them to exceed the axle torque rating. I've had my 2400 lb MF1655 digging holes with the chained rear wheels at 2/3 throttle with the front wheels in the air while digging into a pile of gravel. No big torque load there. Spinning tires on snow takes even less. Now pushing a 7000 lb trailer up hill, with 800 lb of buddies sitting on the back of the tractor, at full throttle, and not spinning results in serious torque loading. Something like when you moved that house. Somehow, I don't think the GT pulling tractors could do that job. Not enough weight to generate the required torque.
I don't know about 10000 lb, but they will certainly carry more than the manufacturer recommends as a maximum. A bigger concern than snapping the axle is bending the axle. When new, the manufacturer had to pay for warranty work if the axle failed, therefore the recommended max load rating is partially set by his willingness to make good on the warranty as well as the actual strength designed in.
However, that tractor hasn't been new for quite some time and wear and tear will have taken at least some toll on the drive train and no one knows how it was treated along the way.
I have no problem overloading components which are inexpensive to purchase, readily available, or easily repaired in-house. Since the OP lives in the UK, parts are not inexpensive, definitely not readily available, and, unless he has access to a well equipped machine shop, not always easily repaired. Not everyone has Kevin Beitz's talents and skills to create working masterpieces out of scrap, which is why I tend to give conservative advise.
Would I put a fork lift on the rear? Absolutely! Carry a 250 kilo payload? Of course! Carry a 500 kilo lb payload? Hmm, maaybeee. Carry a 1000 kilo payload? That would require at least 450 kilos of counter weight on the front end in order to maintain steering. Does the OP really want to drive around with a constant 450 kilo load of weights (24 42 lb suitcase weights) on the front axle while doing other chores, or alternatively, put them on and take them off as the need arises. Power steering becomes kind of necessary at that level of payload and counterweighting.
The stability of a fork lift on the rear is undeniable. The repair cost should something fail in the rear end will be high. This is not a utility tractor that has the capability and mass to deal with 1 tonne loads. It is a garden tractor, or, more precisely, an estate tractor (Wheelhorse's designation, not mine), and has neither the capability nor the mass to play at that level without incurring damage at some point.
If the OP is set on using this attachment, mount it on the front where reinforcement and modification is easier and cheaper and any failures can be dealt with easily and in a timely manner with less stress on the pocketbook. Power steering is a must for working at these weights, no matter which end is used. Some of the necessary counter weight can be left in place without interfering with other tasks and additional counter weight can be easily hooked up to, or dropped off from, the 3PH as needed.
My MF1655 with a FEL will pick up and transport over 700 kilos (over 1500 lb) on a hard surface using only the counterweighting that it is normally equipped with and which did not include wheel weights. The FEL is a bit further out front than a dedicated fork lift is normally so that a 1000 kilo lift is possible with appropriate reinforcement and larger tires than what it has. That was a one time only lift that was actually beyond the limits of the tractor as it was equipped. It couldn't climb the 40 mm (1.6") lip at the garage entrance.
Bottom line, a 1000 kilo lift is a high risk endeavour. I'm not saying that it can't be done, only that you should expect to break it at some point if you plan on making a habit of lifting in that domain. Forewarned, and all that rubbish.
p.s. Fabricating a truss for additional support may help the axles. What effect will the maximum forces have on the differential carrier and gear meshing either with or without the truss? Something else to think about.