I have noticed that the kind of customer who wants me to use his own supplies is the kind that will expect me to work for nothing, and probably want me do drive out to his place to price it out , as well as help him tow it in for free, too.
I won't use customer supplied material, period, unless they want to go to my jobber and buy what I specify over the counter. So far, nobody has taken me up on the offer. I think one guy went and priced it out, then came back and gave me the OK to do it my way, which actually saved him money since I have the required materials on hand and not by the quart or whatever.
It's just not worth the risk. For starters, it will be an unfamiliar material, which increases the chance for error, as well as usually being the bottom of the barrel quality-wise, which is not helpful either. And then guess who gets to be responsible for quality issues? The customer, or the poor guy who caved in and agreed to do something against his better judgement?