I don't know whether "it's cool or uncool" ... I just know what I'd do/what I did ...
When asked by some EOD guys at an airshow @ 10 years back to buy an EOD ballcap and/or T-shirt ... I demurred.
I said: "No thanks, guys ... I haven't EARNED it. I'm certain you wouldn't wear Naval Aviator Wings for the same reason".
A4sForever has it right on the nose. If you didn't earn it, don't wear it. I really get chapped seeing guys wearing military garb, flight jackets usually, with wings and squadron patches when it is obvious they never served a day, shades of old Skeeter who went down in flames (Bravo Zulu, A4'sForever). Ball caps and T shirts are different in my opinion as they don't display your name along with the unit. I wouldn't feel comfortable in one but I don't think it is flying false colors. I have a ball cap that displays the U S Equestrian Team logo and I wear it in support of the organization but I don't think anyone is going to mistake me for a competitor (I'm 65) nor would I represent myself as one.