Flight Instructor reported taking evasive action to avoid a conflict resulted in an airspace violation and a NMAC event.

Narrative

Me and my student were cleared to takeoff from runway XX. Before taking off we requested a Southwest VFR departure; the ATC instructed us to turn right to the south and proceed on course to our destination when reaching 2000 ft MSL. Bravo airspace is at 2000 ft MSL; so we had to stay at 1900 ft until cleared it. When we were at 1900 ft MSL heading south; (under the 2000 ft bravo floor); tower told us that a helicopter was going to pass in front of us at the same altitude from right to left inside the D airspace; heading East. I saw it and it was closer than I'd like in order for me to feel comfortable. Behind us there was another aircraft; and below there was other aircraft in the pattern; so the safest option was to climb. We climbed to approximately 2020 ft to 2050 ft MSL; crossing the bravo airspace boundaries momentarily. Once we cleared the conflict we descended outside of bravo airspace and continued the flight without further issues. Since the helicopter appeared to be in intercepting course to us and there was other aircraft below and behind us; I decided the best option was to climb to clear the helicopter. I felt this course of action was appropriate; but next time if anything similar happens again; I could request a bravo clearance when ATC issue me the traffic alert; request more information from the controller or request a different heading.