Air carrier flight crew reported two separate and conflicting RA alerts during departure climb.

Narrative

I was the Captain; and Pilot Monitoring" of Aircraft X from ONT to ZZZ; on (date). We departed on the RAJEE3 SID and were level at 7000 ft. between RAJEE and ELLBO (if I recall correctly). We had been told to maintain 7000 ft; for traffic. We then received a TCAS RA to climb; for small aircraft traffic in front of us. (Note: we never received a TCAS "traffic" alert; it went straight to a TCAS RA to 'climb'). My First Officer; as Pilot Flying; immediately disconnected the autopilot and began climbing; following the flight guidance TCAS pitch bars. I keyed the mic and advised ATC that we were climbing on a TCAS alert. Air Traffic Control responded "Negative; Company Aircraft X; maintain 7000 ft". I then responded back; "I have to follow TCAS instructions". After beginning the climb as per TCAS guidance; we immediately received TCAS audible instructions to "Descend"; for traffic above us. My First Officer then began to follow pitch guidance for the "descend" RA. During the descend RA instructions; we received audible "monitor Vertical Speed" instructions; to which my First Officer leveled out to stay within the pitch commands. We had been sandwiched between two general aviation aircraft; receiving conflicting TCAS RA commands. After receiving a "clear of conflict" audible from the TCAS system; we returned to our assigned altitude of 7000 ft. We continued our departure following ATC instructions with no further conflict. I believe we may have come to within 400 ft. of another aircraft in flight and I am deeply concerned with the position we were put in by Air Traffic Control; as well as the instructions given by an Air Traffic Controller for me to ignore not only my onboard TCAS audible alerts; but my Airline Flight Manual's Limitations section instructions as well. I fear this report will land nowhere; and be covered up with mounds of paperwork. While the Air Traffic Control system continues to be hampered by budget cuts and restraints; and those responsible go home to their families and the comfort of their homes; we Pilots who deal with these shortcomings; may not go home at all. But what do you care? In my XX years of flying; over 24;000 hours total time; I have never received a double TCAS RA; until today.Suggestion: Honestly; I'm not sure. I understand our ATC system has become so congested and our Controllers are put in positions that I personally couldn't imagine; the only solution I see that could have prevented our event would be to hold our flight on the ground longer while the GA traffic was moved clear of our SID. After a commercial aircraft is airborne (burning expensive jet fuel); I believe private GA aircraft should be turned; climbed; or descended out of the way of a departure/arrival corridor."