Air Carrier First Officer reported encountering rough weather on final approach and continuing to landing after receiving a 'pull up; terrain' warning. The FO stated the crew continued because the runway and PAPI were insight.

Narrative

On approach to Runway XXR at ZZZ; we experienced gusty conditions during the approach. Due to the gusty conditions; we configured the aircraft to land with flaps 30 and a reference speed of 161 knots to compensate for conditions. I disconnected the autopilot and auto-throttles around 3000 feet; and during the entire approach; we experienced plus and minus 10 knots of gains and losses of speed. We continued on the approach; and there was about two times where after correcting the thrust setting due to increase of speed to around 172 knots; we experienced additional gust gains after reducing thrust that caused us to exceed the flaps limit of 175 knots by three knots to about 178 knots. The flap limit speed was exceeded; and the over speed lasted about 2-5 seconds as I reduced the thrust in order to reduce the speed to our approach speed of 161 knots. We continued down on the approach; and at minimums of 512 feet barometric; I transitioned to be below the glide slope by using the PAPI system. Around 200 feet and below; we received an aural warning of 'Too low; terrain;' the Pilot Monitoring said to continue since we were in visual conditions and terrain wasn't a factor. We landed safely while within the touchdown and taxied to our gate. Maintenance write up was logged for flap limit exceeding and maintenance was also contacted to inspect the aircraft.