So actually this won't work. An attacker can just add an input with a public key that is the multiplicative inverse of your public key.
So if A = y and B = (1/y) the signature is verified as:
g^m = (A*B)^r * r^s
g^m = (y/y)^r * r^s
g^m = 1^r * r^s
g^m = r^s
which is trivially forged.
Doing a little more research I believe it is possible to use elliptic-curve arithmetic to add the private keys together to create a "master" private key for signing and this master private key can be verified by adding the public keys together to form a master public key.
And in elliptic curve arithmetic, you add the inverse, yielding the identity element. An attacker can then add their own public key to avoid the point at infinity. The combined "master" public key is insecure. So this doesn't work.