B737 Captain reported a low altitude warning at 1000 feet from the aircraft and ATC when they descended below glide path in response to a TCAS/RA.

Narrative

Cleared for the ZZZZZ to ZZZ Rwy XXL; configured for a 30 flap landing. Prior to intercepting final approach course; we were advised of traffic cleared for the visual to XXR; call sign Aircraft Y. Aircraft Y reported us in sight and was told to maintain visual separation with our flight. Turning final; we were switched to tower and cleared to land with PF hand flying. However; Aircraft Y had a faster approach speed than us and was gaining on us from our right rear and with a descent rate that was closing with our descent path. Passing approximately 1000 feet we received off and on TCAS TA's. We received a TCAS RA with the descending traffic 200 feet above us and closing; at that point PF executed TCAS RA; rate of descent increased to maximum of 900FPM. No vertical speed/sink aural alerts sounded.PM reported TCAS RA descent to tower. Close to 500'; clear of conflict and monitor vertical speed aural alerts sounded. A nose up attitude was commanded. In landing configuration; this was maybe 3-4 nose up; with a descent rate of 300-400fpm. VSI showed three red and one white at this point. PM reported clear of conflict. Aircraft 500' aural alerts sounded. PF replied stable with rate of descent returning to approximately 700FPM and on glide path. Tower asked our intentions. We replied landing. At this point Aircraft Y was in front of and above us for XXR and we were able to maintain visual separation. Tower reported low altitude alert and alt setting XX.XX; which we confirmed and recleared us to land at approximately 400'. Landing was uneventful. FM 8.150.10.38 INHIBITS section indicates that this sequence should not have been able to happen; as descent RA's should be inhibited below 1100' radio altitude. I have not submitted a report without at least one error listed. During our debrief; which we included a jumpseater (different airframe); we could not identify any errors; but we may have missed something.