The mold remediation at this time is a moot point as I caved and hired the company they insisted on using which has cost me $6k more then a company of my choosing would have done it for.
Ouch. I think this puts you in a difficult position. Seems to me either you should be in charge of who you hire to make good with your customer, and part of that is that if it doesn't fix the problem then you are still on the hook to repair the repair, as it were. Or the customer should directly hire the contractor he wants to hire, and then get you to reimburse him for the cost of the repairs. The advantage of that is that if his contractor doesn't fix the problems, you are clear of the situation and he has to sue that contractor himself.
So hopefully you can trust this mold remediation company to do that job well. If you aren't sure, then maybe consider hiring a different company altogether - ideally one with a great reputation in mold remediation. The fact that they are trying to get work away from you by going over your head suggests they aren't as trustworthy/ethical as you'd like.
I guess in your shoes I'd either get the customer to agree to give you the chance to make good on your terms (you pick the providers and take full responsibility for the outcome) or allow the greedy mold remediation company to take over all the work on condition that the customer signs away his right to sue you if the greedy mold remediation company fails to fix everything to the customer's satisfaction for the agreed-in-advance, bearable-for-you, price.
Option #1 sounds better to me, but requires you to figure out how to charm the customer into it, or to get a lawyer to explain to the customer why he has no choice other than to give you that chance.