Always worth remembering when someone is asking you to either work for free/low fees because they have a ‘budget’, is to remember that you have a budget too. Your budget is your rent, your electricity, your food and so on, and your budget is not any less important than theirs.
I’ve used this analogy before but you don’t go into a supermarket and say hey, I’m working to a budget, so I’ll give you X amount for this packet of biscuits. If a company doesn’t have the budget or can’t work their maths out, that’s a them problem, not a you problem.
And lastly, if someone says they have a budget and it’s too low for you, that’s fine. A budget is a budget. The problem comes when you feel pressured to under-cut or under-value yourself at the expense of your own finances because someone else doesn’t have the money to pay you.