What is the procedure for going to another branch's CCC?

The first thing you need to do is to talk to your state headquarters to see if they will fund it and if there are slots available as an Engineer Officer, so the answer to the last part is yes, if you attend the Engineer CCC, your primary Branch will now be Engineer. When I moved to Kentucky, the State HQ asked if I would branch transfer since they were short of company grade Engineer Officers, I was a senior Infantry LT at the time. I told them yes, they transferred me and enrolled me in what was then Engineer Officer Advanced Course. Completion of the course makes you a qualified Engineer Officer.

Going to a different CCC adds a branch, it switches your primary AOC but you still retain both.
You are already in an EN BN, going to EN CCC requires you to be in a 12A duty position.
Talk to your CO and your BN XO about making that move.

I was the same as you. When I graduated OCS, I went to the Maine Army National Guard Group HQ and asked to become an Engineer officer. The XO asked if I had any engineer experience and I said no, but I had plenty of math and science, BS Microbiology minor Biochemistry. He said he needed a Chemical officer, and I said fine, what's that. I did my Chemical Officer Basic at Ft McClellan and was the Group Chemical Officer for four years. I realized that if I wanted to advance, I needed to become an Engineer since we were an Engineer State at the time. I spoke with the Group S3 and Commander and they agreed and arranged for me to go to Engineer Advance Course at Ft Leonard Wood, first class offered. I continued as an Engineer officer until my retirement. You should talk with your leadership about your career concerns. They should be able to help you or point you in the right direction.