I have typed out and then deleted three separate responses to this post.
For now, I will leave you with three questions:
1. You are aware that, on top of the 500$ stud fee, artificial insemination will cost an additional 4-500$ from your vet? You can not do it at home with a turkey baster. You have to test your mare's estro levels and test ova maturity sizes to pinpoint your nine hour ovulation window, and correspond with the stallion owner to have fresh cooled semen collected and shipped to arrive to you within that window.
2. You are already limiting your potential foal's possible marketability by breeding a mare that I assume has not proven herself or made a name for her self in any working activities/showing or competition. Why would you further wish to diminish any value of your foal (and essentially make it worthless to anyone but you) by "not caring what breed the stallion is as long as he is under 500$". It's nice to imagine you will always keep all your horses, but part of being mature is accepting that things sometimes happen in life that are beyond your control, and sometimes sacrifices have to be made. I assure you, if the decision came down to whose mouth to feed, yours, or the foals, your father would choose yours in a heartbeat. So by intentionally breeding a foal where you literally don't give a **** about it's parentage, you are creating a foal anyone besides you literally won't give a **** about.
3. Color should be the LAST thing on your list when looking at a stallion - behind health, conformation, temperament, talent/abilities, marketability, etc. COST should be well down on that list as well.
You are going about this whole thing very backwards. I beg you to rethink your approach and do a LOT more research before deciding whether or not you even SHOULD breed your mare, and should you decide that AFTER you have done your homework, you should - THEN start looking for a stallion.
If you just want a bay filly that looks kinda like the mare - buy one. There are bay fillies all over Craigslist for about 150 bucks - far less than you would invest in vet care, mare care, AI fees and stud fees.