Air Carrier Captain reported an NMAC event during initial climb. Flight crew complied with ATC instructions and subsequently followed RA directions for a safe resolution.

Narrative

On initial climbout of ZZZ; we were instructed to expedite to 3000 ft. Upon handoff to another Controller we were given a climb to 6000 ft. About that time ATC pointed out traffic and said he wasn't talking to that aircraft. As we were climbing rapidly through approximately 3200 ft.; ATC instructed us to maintain our current altitude for traffic. The First Officer (FO) pitched the aircraft to level and return to 3200 ft. Of course this takes some time when climbing as rapidly as we were (light aircraft; previously instructed to expedite). As the aircraft was leveling; we received a CLIMB RA. Before I could process that; we received a Maintain Vertical Speed RA. The RA was complied with. We passed under the threat aircraft by 300 ft. In that maneuvering; the airspeed exceed the maximum per the regulations for flight below Class B Airspace. In reviewing the report page prior to going to ZZZ we knew there was a lot GA traffic in the area. Some of the narratives are eye opening. Given the history at that airport; this experience; and what Pilots are taught about risk management; it seems a big picture solution needs to happen. I won't pretend to have that answer.