Who is your recruiter's "contact"? The link above directs you the CEC Accessions page. If you click 'contact us' you will be directed to a page that lists recruiters and accessions officers based on your location. If your recruiter is not talking to a CEC Accessions Officer, then direct him to the right person, if he is, then the answer you received is probably the final answer. In this case, however, you might want to talk to the accessions officer just to make sure he understands your exact situation.
There are many Lat Transfers in the CEC (mostly SWO's and Aviators). I don't know how competitive it is, but generally you need to wait until the end of you committment to apply for the transfer (4 yrs for SWO, 6-8 yrs for aviator). Technically, it is possible to apply for a Lat Transfer 24 months after commissioning, but this requires your current community to release you (I don't see this happening with the SWO community).
Your biggest obstacle, however, is the accredidation. If they won't accept your degree now, there is no reason to expect that the will accept it as a Lateral Transfer.
I was in the CEC BDCP program several years ago and then actually Lat Transferred out of the CEC to Aviation (at that time, the first person to transfer out of the CEC in over 15 years). The CEC is one of the best kept secrets in the Navy. Good Luck.