...mower starts to slow down and then stops...tractor also slows down...really feels like it is struggling with the weight of the mower combined with the hill.
Mower stops but tractor still running ?....either a clutch on the mower , a belt on the mower or the pto clutch on the tractor must be slipping. ..I think most likely an issue on the mower as otherwise you would have probably had issues with the haybob and baler stopping too.
Mower is perfectly suited to your tractor, it is a fault as noted above you need to resolve and renew your blades.
Regarding loss of traction with the baler, solution is you need more weight on back wheels to aid traction. This is one of the problems with a baler which is quite heavy but is balanced and hence doesn't put a lot of weight on the back wheels of the tractor, unlike a proper tractor trailer which increases weight on back wheels the more you load it.
So what could you do. .?
1. Get rid of those weights off the front of the tractor when towing your baler they just transfer more weight onto the front wheels increasing rolling resistance whilst also transferring weight off the back wheels decreasing traction. A double whammy of negative effects.
1a. Check rear tyre pressures run at lowest recommended pressure to increase footprint.
2. Increase weight on back wheels . 2 methods I can think of. Buy rear wheel weights which bolt on, or fill rear tyres with water/antifreeze mixture.
3.another possibility would be to buy some wider rear wheels...and add weight by one of the above methods.
You appear to have a belting 135 there which you bought well. Changing It for another tractor is fraught with risk so my advice would be try some of the above first.
Your other option without going to 4 wheel drive would be to change to the slightly larger 165 , 265, or 565, all essentially the same tractors under the skin. This will give you more power and more traction due to greater weight. You should be able to get a good one as a straight swap. You need to make sure you get one with power steering and do not under any circumstances buy one with multi-power transmission , they are not suited to hilly terrain.
4 wheel drive is obviously better for hilly terrain, but comes At a price and is a whole load more bits to go wrong. You got by with your 135 as it is so it isn't far off the mark, a few cheap mods and I believe all round it will serve you better longer and cheaper. You may have to drop a gear going up hill...but what of it ? It's not like you've got 500 acres to do ! Points 1 and 1a may be enough.
When you are baling this year you need to shorten the chains of the bale chute at the back of your baler so it sits level instead of dropping like in your photo. It's purpose is to support the bale level as it comes out of the chamber, keeping the resistance constant so you get an even density bale that isn't bent.