You clearly understand, if the single member is Robert E. Lee, and the customer is Confederate Plumbing, LLC then you are going to get a "mismatch" notice on CTR filing and perhaps on 1099 reporting if you use Robert's SSN.
The fault is not yours, it's theirs. Personally, I would not go to any great lengths to fix it.
There have been a number of suggestions made, mostly in the BSA forum. Yours, listing the name of the single member in that capacity, is as good as anyone's. However, when I think through he operational issues and the confusion that might cause, I'll just go with the mismatches...
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.