IMO, if you want to work remotely, then one of the things you have to accept is that the employer is likely to pay you less. Otherwise, they’d be paying local salaries to local employees, and you wouldn’t get the job.

Now, in this case, since they already recognize the cost of living in NYC is different from the rest of the country then you could make the case that you should be making near NYC level of money, because DC is also extremely expensive. But I think that’s a real uphill battle, and I think you’d be likely to lose the offer.

A good employer should be willing to have that conversation with you, but good employers are more rare than hen’s teeth.

So, I think you have to decide if you want to work remotely and get remote-work scale pay, or if you want to get DC-scale pay.