I was in a similar spot as a manager for years. You can't fix everything but you can improve the conditions for your workers and in my experience, this also was the key to making our store the top in our entire state. They will not always ingratiate you with the higher-ups, but if metrics improve, they will notice and at the bare minimum tolerate you.

A few good things you can do:

Write glowing recommendations and beg for raises every year. Also make a point to do mentorship to help each worker build their career and skills in the ways they want. You don't always have to put all of this in a review, but if it looks good and they make progress, do it.
Spend that team building budget every month. Tell your team what it is and ask for their feedback on how they want to use it.
Ask for input from your team and listen to what they have to say. Be transparent about what is going on, what your goals are, and what limitations you have, then try to find a way to find a place of compromise between those limitations and the feedback your team gives. Again, be open about this and ask what they think of any compromises. They'll often have a better idea of what will work to improve conditions more than you will, particularly if you are open with them about goals and limitations.
If it's reasonable and safe for you to do so, encourage them to organize themselves into a union. You probably cannot join this, but you don't need to tell your bosses they are organizing until they have done it. You can act as a screen to their efforts until they have the power to demand recognition.