The buyer seems really opposed to using an FFL and the only reason I can think that someone would be so opposed is because they’re trying to do some shady. That’s just my take.
I can’t speak to NY laws on the matter, but I can definitely understand someone with a C&R FFL not wanting to use another FFL for the transfer. It doesn’t necessarily mean that the buyer is doing something shady. At a minimum involving another FFL is added time and expense for the buyer to go pick up the gun, and any savings on shipping may be a wash (or even a loss) depending on the receiving FFL’s transfer fees.
As an example, I’m pretty centrally located in my city, but it takes at minimum an hour and a half for me to drive to my FFL, transfer a gun, and get back to whatever I was doing. And that’s a no chatting, no browsing, “I’m on my lunch break, let’s get this done” kind of thing. Say you save $30 on shipping by using a FFL. Cheapest transfer I’ve found is $25 (which seems to be about the normal “good deal” transfer price) so that’s only $5 less expensive. Then you take into account the gas and time of actually going to pick up the gun and you’ve actually lost money compared to just paying the more expensive shipping, for something that could have just been shipped to the buyer’s house (again, not accounting for NY laws).
And none of that even begins to address the current pandemic stuff. The buyer could be in a high risk demographic (age, comorbidities, compromised immune system, or living with someone who is high risk) and have a very good reason for not wanting to leave the house (to say nothing of all the “shelter in place” orders across various states right now).
Ultimately as the seller it is your decision. I’m not saying that you’re wrong, and NY law may not allow a C&R to receive the gun anyway. But for a $200 pistol that is sold in multiple places as C&R and given everything else that is going on right now, I can understand a buyer with a C&R FFL being opposed to having to go to another FFL to handle the transaction for a number of reasons, none of which imply that the buyer is trying to do something illegal or shady.