Do I have any leverage for choosing my DS assignment?

No, the extension your Career Counselor does has no effect on your assignment. Also, your profile says USAR but this sounds like an Active Duty question. If you're USAR it's completely different

My suggestion is ask ... they’re not going to kill you.

Also, “I Scratch your back, if You Scratch my back”, will not work. Your evaluations and performance will take you a long way.

Good luck ... Hooah.

The time for leverage has come and gone. If I understand correctly you are already tagged. This is an old method used for filling high demand positions like recruiting/drill field, etc.
They use it all the time in the Marine Corps. If you aren’t proactively working with your respective MOS monitor and pursuing/pending orders eventually the special duty screening team will flag you as a candidate. They generally look for folks wrapping up their first contract and who have nothing set up

They have you cornered in a way because if you stay in they already know where they are sending you. If you want to go somewhere specific you can always ask. Just understand you won’t be seen as a volunteer and may say no to what you request