My Craftsman I used for plowing, when I first built it up, I locked the transaxle, put a gas pedal on, a 110v starter so I wouldn't have to worry about a battery in the winter time, a pull start for when I wasn't near the house, and extended the rear bumper to hold weights on which actually helped me from rolling it backwards when I did wheelies on it and made some nasty divots in the grass in the summer time, and chains for the rear tires. I used it for plowing for like 3 winters I think with this setup and was pushing snow over 1 foot high.
Anyways, there was good points for locking the transaxle and bad points too. Even though it was just snow, ice, grass, and pavement, I think it would benefit this topic.
The good points on locking the diff- never got stuck, even just driving through over a foot of snow not plowing. I would just idle the engine high and let the rear tires dig till it would move. I would push a huge pile of snow with ease with no slipping and stab it into a pile. It would stall the motor from straining it first before it would loose traction and that was with chains on the turf savers in the snow and ice. In the summer, I would take the plow off and do wheelies the whole length of the yard dragging the bumper extension on the ground. And yes it had the stock pulleys too. I had the governor still hooked up with the gas pedal so I could set the engine speed with the hand throttle and use the gas pedal to speed it up when needed. Came in handy plowing and pushing into a snow pile, right before hitting the pile I would floor it for extra speed and then hit the clutch so it wouldn't stall the motor when I stopped.
Bad points on locking the diff-Steering sucks, plain and simple. Because all the weight is in the back, since the tires are locked, all the mower wants to do is go straight. You can combat this with better tires on the front to dig in so you can turn better but will never be the same as turning with an open diff. You also put an enormous amount of twisting force on the axles just turning and if you got a good set of rear tires in the back digging in then you probably will break an axle or something by just turning. You can turn a little with no problems but sharp turns you can feel stuff binding which is where premature wear happens that leads to breakage. locking the front axle from pivoting helps a little also, in an attempt to get the frame to shift the weight to one side for turning by using the front tires to get the inside rear tire to slip but most frames will flex anyways and might not help.
My transaxle that I locked was a 3 speed with 3/4 axles. I was trying to pull a car and broke an axle doing it. Probably not the best move but I had to pull the transaxle and put a different one in. I put an F/N/R in because it bolted right up with a 4 inch pulley on it to get some kind of speed out of it for plowing and didn't lock it. Kept an open diff, kept the turf savers with the chains and used it for 2 more seasons of plowing. Worked just as good with the open diff and since it was a F/N/R was way easier to plow and turn too. I thought about locking it before I put it in but hated the issue of turning and didn't lock it. I never drove in snow over a foot because I knew I probably wouldn't make it through, instead I just dropped the plow and plowed through it. It worked out great in the end. I still got this thing but never dug it out of the shed for another season of plowing, too lazy I guess.
I would look into individually breaking each rear wheel with handles or hand brakes to give you manual traction control. If one rear tire is spinning then brake that tire so the other rear tire will take the power. A little tip from the big boy tractors.
If you want to lock the rear, be prepared to sacrifice your steering. In the end, you'll probably be on hard and soft ground most of the time and not being able to steer good sucks.