I immediately thought of the Bumbleride Era, but that might not fit the budget and it also isn't available everywhere. Still, I'd check if, by any chance, you can't put your hands on it...
My next suggestion would actually be even more expensive, but I'd get it pre-owned - a Bugaboo Fox, or even better, an older Bugaboo Buffalo.
From the more affordable ones, the Cybex Talos S is definitely a reversible option with large, all-terrain wheels, and the ABC Design Salsa 4 Air.
I can also recommend the Kinderkraft Everyday (full Kinderkraft Everyday stroller review here) for its spacious parts with a parent-facing possibility, and large wheels.
Lastly, I'll mention the Baby Jogger City Premier, as that is one very nicely made reversible pram, BUT it is not sold much anymore. Just check, perhaps...
I know I didn't go much into the truly budget-friendly strollers, but I am trying to advise you with my conscience truly in, and I wouldn't go for a cheap cheap reversible all-terrain, as that would be not really enough all-terrain, or the wheels would start to rattle very soon... or the seat would be super small. I would rather suggest getting a pre-owned model if your budget is tight.
Perhaps, if you have something else in mind, send me your considerations and I'll gladly talk with you about them ^-^
All the best!