He's meant to be marginally older than Tony in the show. He was initially with Johnny as you said. I feel this is why he is sent.
It was my understanding that Johnny and Junior worked together when possible although they ran separate crews. My assumption was that most of Johhny's guys were okay with Junior and his guys and vice versa of Junior's crew w/ Johnny. I would imagine there were a few guys in each crew generally disliked, but as far as in family cooperation, you can safely assume that Johnny and Junior told their guys avoid issues with each other.

Pussy seemed to earn well while being a generally well liked and well included person. He was friendly with Johnny Sac while never rising above the rank of solider. To be that trusted by Tony, one would imagine he was well liked by Johnny as well especially after the 'Unrest of '83". This is what leads me to believe that Junior would sooner see Pussy as a neutral messenger than see Tony that way.

Another easier answer is that Tony stayed in Jersey for his own safety, why go risk getting shot at by Junior's bodyguards if he didn't really have to? If you were Jackie and were about to bump Tony up, why send him when you can send a career soldier like Pussy just in case another goes wrong.