I’m a registered landscape architect in Pittsburgh, PA and owner of a landscape architectural design-build firm. It sounds like you did everything you should have and performed in a completely professional manner. As difficult as these situations can be, try not to engage this person in any further discussion or debate. It sounds like this guy was told he would have weeds no matter what, as any good pro would, and is just using threats to get something for nothing (free maintenance). If he is that cheap, he will probably give up when his legal council tells him that he has no case or wants to charge a legal fee. If it goes further (a big if !) it will be laughed out of court.
I wonder if this is client creates these kinds of problems for other service providers that have worked for him. How well do you know the installer? – where you on-site to know for sure if he applied preemergent under the mulch? If preemergent was not installed – this can happen with unsupervised crews in design-bid and design-build. If the preemergent was installed, the client might be exagerating his claim.
Has your client only expected the fantasy of no weeds and not have been difficult in any other way? Have we all run into one or several clients that have a distrusting, “expect it all” mentality? – great design for a low fee, top quality installation from the low bid, specs that will control and take responsibility for unsupervised installers, the right to renegotiate or not pay a final payment – payment invoices for lump sum contracts! (I just provide receipts of their payments to me), asking for freebies and the ability to control nature with zero maintenance and no weeds.