Is it possible to split a key for transaction signing without using multisig?
There is a way for you to do so, but it involves something called Threshold Signatures which lets you do most of what you want (n-of-n without multisig). The thing about threshold signatures is that they require at least 2 round trips. That is, A and B have to exchange partial signatures a few times before the signature is valid. For 3-of-3 and higher, you need 3 trips to each person in order to produce a valid sig. Also, threshold signatures haven't really been implemented in any software than the one the Princeton researchers made.

The other method of sort-of doing something like what you want is using Shamir's Secret Sharing. I know Armory implements this for fragmented backups but you can't use the sharded pieces for actually signing, only for recovering the actual private key.