B737 Captain reported a bounced landing that resulted in a tail strike. The aircraft tail skid was damaged and a maintenance inspection was requested.

Narrative

This was the final leg of a 2-day; 3-leg pairing. The first day had gone off with no issues. The final leg; ZZZZ-ZZZ; was uneventful with a standard departure; climb and cruise phase. The First Officer (FO) was pilot flying (PF) while I was pilot monitoring (PM). Weather app was showing some light to moderate turbulence during the descent and we expected weather to begin building around our ETA; so we coordinated with the flight attendants (FAs) to be seated early.We initially set up for and briefed the RNAV (RNP) Z Runway XX into ZZZ due to the ATIS; recent runway usage and other aircraft being assigned the same. ZZZ issued a descend via clearance on the ZZZZZ landing south; so we re-briefed and set up for the RNAV (RNP) Z Runway XXR. The remainder of the arrival phase was uneventful: we joined the approach at ZZZZZ1; maintained the ATC assigned speeds; and the weather seemed surprisingly calm for ZZZ in the summer.Around 250' AFE we received a TCAS TA either for a Runway XY departure (who we were crossing over) or an aircraft taxiing on 1/2. We verbalized the alert; saw the traffic; and continued.Around 70' AGL we experienced a strong decreasing performance downdraft. As I began to verbalize the need to add power; the FO began increasing power to compensate. I did not see the airspeed decrease below Vapp or Vref and based on my (admittedly limited) experience on the 737 thought we had adequate energy. However; the additional power was maintained later than usual and the aircraft bounced. This produced a significant startle effect and triggered the takeoff configuration warning which added to the confusion. Reflectively; the FO retarded the thrust levers and the aircraft began to settle. It took a second or two for us to both wrap our head around the aircraft's state and the FO asked if we should go around. At this point I saw a pitch attitude of approximately 7.5 degrees nose up and felt that we were only a few feet above the runway. My concern was that adding power would produce a nose up moment and drive the tail into the runway; particularly with the slow spool-up times of the LEAP engines. I felt that we were too late to successfully go-around from such a low-energy state without a serious risk of aircraft damage and stated that we should continue the landing. The rollout was more disorganized than normal but manual braking and reverse thrust was accomplished.We exited the runway and taxied to the gate. I was concerned with the potential for a tail strike; having felt a very slight; barely perceivable vibration during the touchdown that I was hoping was just a main wheel spinning up. The parking checklist was accomplished.The FO and I conducted a debrief of the energy management on the approach and landing. We discussed the need to immediately go-around with that loss of performance and bounce; particularly since the engines were still spooled up at that point. We debriefed why I felt landing was the more appropriate choice at the time we had both caught up mentally to the aircraft's state and how disorienting the bounce was. We also discussed how despite being clearly at the limit for pitch before tail contact; the nose did not visibly appear to be that high; especially compared to other aircraft we had flown before that did not have such serious tail strike concerns.A post-flight inspection revealed damage to the tail skid bumper but no visible damage to the aircraft fuselage or structure. The tail skid indicator still had a small amount of green remaining. We submitted an ELB report and notified Maintenance.