ZLA Center Controller reported an aircraft descended below the Minimum IFR Altitude in response to a TCAS/RA for an unidentified VFR target.

Narrative

Aircraft X was on an IFR flight plan to ZZZ at 090 Routed JLI.v514.TRM..ZZZ. Aircraft X was approaching TRM and ready to be shipped to SOCAL Tracon Sector X. Sector X calls and requested the aircraft to be on a heading of 360 for an unidentified VFR target they indicated was in a descent. This VFR target was opposite direction of Aircraft X at 095. I issued the heading to Aircraft X per Sector X request; but immediately afterwards referenced my MIA's (Minimum IFR Altitude) in the area and re-issued a 020 heading. (Aircraft X was currently on the victor airway in an MIA block of 108 going into a MIA block of 090 and further lower after that. V514/V208 support an altitude of 090 only on the airway.) The original vector of 360 requested by Sector X would have put the aircraft in violation of that 108 MIA block. Once i issued the heading of 020; I did a quick scan of my sector. Seconds later; I returned to Aircraft X and issued a traffic call for that unidentified target. Once I finished the traffic call; Aircraft X indicated that they are responding to an RA and descending. I issued a 20 degree right turn to Aircraft X in order to further separate the aircraft and start Aircraft X in the direction of a lower MIA block. I then issued a Low Altitude Alert to Aircraft X informing them of the MIA and when able to climb back to 090. Aircraft X descended to 085 and then said they finished executing the RA and were in a climb back to 090. I then shipped Aircraft X on a heading to Sector X.I could have given Sector X control of the aircraft; shipped Aircraft X; then Sector X could have vectored the aircraft as their MIA's are different than ours EnRoute (I believe since Sector X requested the aircraft on the heading) I was unsure if I was cleared to comply regardless of my MIA's; but I wasn't 100% sure and so I did what I thought was the safest course of action at the time.My regret looking back was issuing an 020 heading instead of a 040 or more. Only after the incident did I realize the 020 was essentially what the aircraft was doing while on the airway. Issuing the 20R vector to Aircraft X in combination with their descent put Aircraft X below the MIA when the turn was initiated. This was likely wrong; but in the moment I put Aircraft X further away from traffic and directed to a lower MIA block. I failed to re-coordinate with Sector X prior to shipping the aircraft as I was caught up in the moment. In my judgement I feel as though I attempted to do what I could to keep the aircraft; and its passengers involved the safest.