How would you suggest I find a CW3 to get a recommendation for flight school?

Start with not using a photo of you in uniform wearing your patrol cap improperly.

Then I would ensure your packet is squared away, as that is one of the easiest ways to get turned down. Find a test date for the SIFT and then backwards plan your studying. You've listed your MOS as a BFV mechanic so I'd assume that you have a Warrant officer within your unit, talk to that person and see if they have any contacts in the aviation world.

Go hang out at the local coffee shop. One will show up eventually.

Best to Start with your unit chain of command.