In fairness to the OP, he said he had a budget, which as it happens is too low, he/she didn't say they want cheap, and recognises that it will be more than.
I still say the best way is to see the job and price it, when you know what work is involved.
A 4bed with zoning and additions to current regs, is going to be £2-2.5k as an average, and without a doubt there's plenty of cowboys and illegals that will take your cash upfront and chuck something in for much less, whether you get all the paperwork and it's safe to use is another story, but they will be long gone and don't care.