Is it possible to reclass after 12 years active service?

It's possible but very unlikely. Most MOS's have time in service limits. Your RCP is 14 years which limits your ability to PCS and attend TDY enroute. It also limits the MOS's you can select because you can't meet the SRR for some of them. On top of that, your MOS is balanced. Normally, you have to be in an over strength MOS to reclass outside of reenlistment, and you must reclass into an under strength MOS. In order for your Career Counselor to help you reclass you will most likely need one or more waivers or exceptions to policy. ETPs are for exceptional circumstances or exceptional people. Being an 11 year E5 in a large MOS like 91B is not indicative of either. Your Career Counselor has a mission to complete and your case will take several hours, spanning several weeks, and will almost certainly be denied by HRC.

Realistically, your best options are 25E, Recruiter, Warrant, or Reserve Component. 25E will require a TIG waiver from the school house and an RCP ETP from HRC that just might get approved since it's a SMAPP MOS. But, again unless you have something exceptional about yourself HRC will very likely deny it.

If you don't want to be a recruiter or a warrant, you can separate into the RC and apply for an AGR position with them.

For starters, what's your In/Out Call look like for rank and MOS and what does the In/Out look like for the MOS you want?