If you think you can get action from the same book I'd just ignore it and eat the $2250 loss. Back in the day when I went on my epic run (hence the name, but now I just suck and bet for fun), had something similar to you happen with the above for around $4kish (end of a 10 week epic c-money type run). I ate the loss then had my bro in law open an account with the same local and just ran it up until I started losing. Funny enough that same local has offered me an account not to long ago as we ran into each other at a bar. Nowadays I would just give my full name and move on as that 3k would be like 20% of my roll for all things gambling.
TLR- If you can make the $2,250 back before they figure out who you are + extra AND get paid, don't give your details and eat the loss now.