I have a dollar store spray bottle, I have apple pie in it for when I want to spray ribs or pork steaks. I have not seen mold in it. It has been in the fridge fil for years, I just refill it. Could be the alcohol keeps it from molding.
My take on it would be to keep using the cheap sprayers if they work. I use Whirl type product when cooking on the griddle, it builds up in the squeeze bottles and I tried for 3 days one time to clean it out and, finally it hit me that this bottle cost me $1 why am I wasting my time trying to clean it.
Only other suggestion would be to put some vinegar in it to try to clean the mold out or keep it from starting.