Hey TJ, I believe I know what your problem is.
The Massey-Ferguson 3pt lift is designed to provide either
simple position control OR
Draft control primarily for plowing. That design and features were still common when I bought my 2011 MF2660. Same mechanisms with the big spring and all that.
What is little-known and not well publicized is this: Having the Draft lever in the full up position DOES NOT really disable the Draft control but merely desensitizes it. Some dealers even tell customers that it does disable Draft Control. I think even the manual says so [maybe the shop manual admits desensitizing rather than disabling?]
ANYWAY, here is the problem you are experiencing:
Above some large amount of weight on the 3 pt hitch, that weight overcomes the desensitized Draft Control and causes the system to be in Draft Control. MF actually sells an override kit to to take the Draft Control totally out of the picture. What that does is: 1) Remove the pin connecting the top link to the "big spring." That big spring controls input to the Draft Control system internal to the tractor. 2) The kit then attaches the top link "beam" (the heavy cast iron one with several holes) to fixed points in behind the area where the top link attaches to your tractor and anchors it so it cannot move nor affect the Draft Control. I'll post a picture here in a minute. I think once you see how that works you can make up your own bypass linkage rather easily. The kit from MF costs around $130.
This is with the kit installed. Note the empty hole up top where the 'beam' used to be connected to the big spring. See cotter key in the new shaft through ears on the rear axle.
View attachment 658721
This below is how it looked BEFORE the kit was installed, Note the pin is in the hole thru the big spring rod.
View attachment 658729
Either you make your own or order the MF kit it will probably be the same on your MF165. Confirm that with your dealer or via my pictures before buying anything.
I think it is good for you to know this no matter what you end up doing. If it were me, I would be more inclined to put some sort of ballast on the back like a heavy bush hog or some weight box full of sand or cement or whatever and use the FEL for moving hay bales. I assume your loader has the strength for that task. The rear ballast will fix your "lifting the rear wheels" disease.