The Toyota is the right tool for the job, so I'm sure it'll be ok

I have an open diff on my Bighorn, and an LSD is definitely on the list of things I need to do. A locking diff would be the dream but they're pretty serious money to get as a retrofit item. I'm liking the idea of an Aussie Locker on the front axle, not sure if you've seen them but they seem like a good solution for the front axle as I'm guessing your truck is like mine and is RWD when on the road?
The best solution for hay is to chuck it on a trailer in hindsight. No mess in your vehicle and you can carry more of the stuff. Even better find a farmer who delivers it then he can wreck his toys carrying it
How about an import pickup

I want this but it's very much out of my budget unfortunately:
One day...