Air carrier Captain reported aural sink rate warnings on final approach to MGGT airport. The Captain continued the approach to landing and elected to not perform a go-around.

Narrative

This report is to document the multiple approach changes that occurred as well as a low altitude sink rate warning on approach into Guatemala City. One hour out from landing; the winds in MGGT were variable at 4. We briefed the ILS Z Rwy 02. The briefing and descent check were completed well before top of descent. Approaching the descent point we noted the ATIS was calling the active runway 20 with winds 120/4; BKN 1800 ft with Towering Cumulus and unrestricted visibility. We completed the brief and set up for RNP Y Rwy 20 prior to top of descent. We requested and MGGT Center cleared us to GT566 (IAF). On initial contact with La Aurora Approach we were cleared direct to KATAS for the VOR Z Rwy 20. We stated we were prepared for the RNP Y and requested clearance for that; but approach said they were unable due to traffic. We briefed and loaded the VOR Z Rwy 20 and proceeded direct to KATAS. We briefed the VOR Z to 20 and were on path for the approach. Approximately 5 minutes from KATAS; we were issued holding instructions with no EFC; however it was clear we were stacked in holding with two aircraft below us at KATAS. We established a bingo for SLP of 7.5 and with 9.5 onboard; we estimated we had at least 20 minutes of holding and all should workout if the preceding aircraft were given clearances without much delay. We notified dispatch; FAs; and passengers and completed 2 turns in holding. Entering the third turn in holding; we were given vectors in a partial box pattern and eventually a greater than ninety degree turn direct to KATAS with clearance for the approach. The vectors complicated the approach unnecessarily and resulted in an overshoot of the inbound course to KATAS and a minor S turn to final.We were fully stable at 1;000 ft with checklist complete; configured with gear/30 flaps; on path and on target speed of 152 knots with the runway in sight. At approximately 500 feet we had a sustained gain in airspeed of about 15 knots. I pulled about two knob widths of power. The airspeed stayed at 160 and we were showing one red on the 3.5 degree PAPI. I attempted to lower the nose to get back to the path and we got our first sink rate warning. We were visual and I made a small correction to raise the nose but stay on a one red; three white PAPI indication to correct the sink rate. I pulled another knob width of power to reduce the airspeed and attempted to lower the nose slightly to regain a 1000 ft aim point and we again got a sink rate warning. I didn't want to reduce power any more in case we lost the added headwind. I corrected again raising the nose slightly and shifted my aim point to 1500 ft. Our speed was holding at 160. We were within stable approach criteria and I continued to landing. We touched down at about 2;500 feet and the rollout and taxi were uneventful.Per FM; for sink rate warnings at or below 500 ft the PF action is to execute a missed approach unless in visual conditions and the Captain determines that the momentary deviation is being immediately and safely corrected. We were in visual conditions the entire final approach. I did feel I was making immediate and safe corrections; and we were still in a safe position to land after the sink rate was corrected. I also knew from experience that the first part of Rwy 20 has a pronounced downslope. The unique slope of runway 20 would partially mitigate the threat of a slightly elevated vertical speed. In addition; I considered the threats of a missed approach in mountainous terrain with an approach controller who was having difficulty managing arrivals. Overall I felt it was safer to continue the approach if a safe; controlled landing could be made in the touchdown zone.