Air carrier Captain reported an NMAC with a GA aircraft that was crossing the final approach course and not in communication with ATC. Captain took evasive action to avoid a collision.

Narrative

I was PM; FO was PF. While on final for 35 into AVL we had gotten told about traffic by Tower on I believe our initial check in. I responded that we were looking. We were unable to get a visual on the traffic however. Then a few moments later Tower came back and she said it was less than a mile do we have it in sight. I only responded with no; as we were trying to find the traffic and Tower responded immediately to cancel approach clearance and to climb to 6000. We started the climb then while established in the climb we got an RA to descend. We started to level off and begin the descent to comply with the RA when I looked out the FO's window and there was an aircraft similar to a Aircraft Y so close that I could see the occupant in the aircraft. It was approaching almost directly perpendicular to us. It was slightly lower than us and also descending. I saw that our guidance was to continue to descend but using my own judgment of what I'm looking at visually I chose to say to the FO; no climb instead he's below us. We changed path and began a climb against the RA guidance. While we began the pitch up to climb above the RA soon changed to a climbing RA. We complied with the climbing RA guidance and safely maneuvered the aircraft away from the conflicting aircraft.It was a difficult decision to change the aircraft path by starting a climb when the RA wanted us to descend. Especially because it is supposed to resolve the conflict. But after making visual contact with the aircraft I knew we needed to climb away from it and I feel like it was that PIC decision that ultimately resulted in the desired outcome of a safe operation. I also understand that the RA will be accurate with other TCAS aircraft; however when GA aircraft do not follow any resolution and make their own decisions; they might conflict. Possibly leveling off and canceling our own approach and requesting to be brought back around once the traffic who was not talking to Approach or Tower was passed the final approach course