Hi--
First, I always nursed my kids to sleep and when they were ready they went to sleep without it. There is no should or should not--- there is only what works and what doesn't work. I, too, am an attachment parenting kind of Mom.
Next, with my oldest we tried sleep training for one day--that's it, one day. I was always against it, but my husband pushed and we tried it. I was a miserable mess and so was my son. And for a couple weeks after that he was scared to go to sleep. So from my experience, yes, sleep training can make the separation anxiety worse. I, too, had a super sensitive child who wouldn't let anyone else hold him.
That said, check out the books by the baby whisperer. She has information that teaches them to sleep without them crying. That might work.
In my situation her stuff did not work because my son was waking up due to severe food allergies. I mention this so you can determine if that might be what is happening. Symptoms of that could include some or all of the following: Dark and or puffy circles under the eyes, any type of rash (including eczema), persistent post-nasal drip, trouble sleeping (because allergic reactions cause the body to release adrenaline), constipation (defined by not going at least once a day) or diarrhea, or some combination of the two, a persistent cough, and lots of comfort nursing (to help their tummy feel better). If you notice any of these symptoms I can help you with what your next steps would be. For us, I finally brought him into bed with us (he started in a co-sleeper as well) and that helped a lot.
Good luck!
J.