Learn Spanish (Spain): ¿Cuánto es el presupuesto del pintor?

Yes, in many contexts presupuesto means "budget" (the amount of money you have available to spend). However, when dealing with professionals like painters, plumbers, or mechanics, it translates to "quote" or "estimate"—the calculation of how much their work will cost.

Why do we use es here instead of cuesta to ask for the price?

You could say ¿Cuánto cuesta el pintor?, but here we are asking about the quote (el presupuesto) itself, which is a number or a total amount. When asking for the total of a bill, quote, or check, Spanish commonly uses the verb ser (as in ¿Cuánto es?).

Could I use cuál instead of cuánto since an English speaker might say "what is the quote?"

No, because you are specifically asking for an amount of money ("how much"), which requires cuánto. If you asked ¿Cuál es el presupuesto?, a Spanish speaker might think you are asking them to physically identify the quote document among several different papers on a desk.