I am doing the same thing. I have been using a Mahindra 6000. I have a bale spear on the back and when stacking I put a bale back there. I have not stacked three high but I load the bales on flat bed trailers two high where the top bale is much higher than stacking 3 high. I handles the job with no problem. Frankly, I don't even put a bale on the back most of the time any more because I have learned how the tractor will respond.
I can tell you that I am ready to move to a power shuttle instead of a manual reverser. I am also going to a cab. I just sold the balance of my hay and loading the trailers this weekend where high temp will be about 25. If you have not tried one, check out the power shuttles. The tractors that I have looked at are the MF 5455 & 5456. Also looking at the TYM 903. These are 75-85 pto hp and weigh in at around 8000lbs. If you do much hay stacking and unstacking, I would without a doubt look at a tractor with this feature. These are 4 cyl. engines. If you want a little smaller size tractor check out the MF 3645, 3cyl. It weighs in at about 7000 lbs. These weights are without loader, so on each one add about 1000lbs. I just drove one of those. Power shuttle works great, but it seems a little small for moving large round bales on hilly ground. If you are flat, probably would work ok. Has 91hp rated engine, but cc are on the light side, which means the tractor will do the job but work harder. New Holland, JD, Kubota and others all make similar type tractors. I would also go with ag tires and make sure you get a tractor that has a good wide base. Of course, these tractors start to get in the $, but in my opinion, if you do this a lot, worth it. Also consider what other uses you will be using it, and the other tractors you may have.