B737 Captain reported a momentary loss of aircraft control and flap overspeed after abandoning approach in convective weather. Crew landed normally after convective weather passed.

Narrative

I was the Captain and pilot flying on flight # from ZZZ1 to ZZZ. During pre-departure review; the First Officer (FO) and I noted that convective weather was forecasted at the time of our arrival in ZZZ. We discussed the associated weather threats and continued to monitor field conditions while enroute.During the flight we encountered numerous areas of moderate turbulence requiring us to change altitudes multiple times in attempt to find a more comfortable ride. Our final cruising altitude was FL240 with continuous light turbulence much of the flight. ATC advised us of holding into ZZZ. The decision was made to slow our speed and holding was lifted prior to our arrival.An hour prior to landing I set-up and briefed the ZZZZZ 2 arrival and ILS to Runway XXL; which was later assigned to us. There were two convective cells near the airport; one to the west and another to the east. Weather to the north and south was generally clear. Due to weather near the airport; ATC gave us vectors and step-down altitudes to the final approach course; which required the use of speed brakes and flaps to meet speed and altitude restrictions. We remained clear of the weather at that point; which was depicted on the radar as heavy to extreme precipitation and areas of significant turbulence. Approximately 20 miles from the airport ATC advised us that there was FOD from another aircraft on Runway XXL and changed us to Runway XXR. The FO switched the FMC programming and we both reviewed the approach differences. Approximately 5 miles prior to intercepting the final approach course and starting to level at 3000 feet with flaps 5 and speed brakes extended; ATC advised us that the winds had shifted 180 degrees and were gusty. We were asked if we wanted the ILS to Runway XYL. I made to decision to switch runways out of concern for tailwind limitations and associated issues in the event of a go-around. ATC then instructed us to turn left to an assigned heading and climb to 6000 feet. The FO input 6000 in the MCP (Mode Control Panel) and selected vertical speed +700 while I adjusted the heading. As we made the turn I observed stationary weather to the east of the airport; while also observing that the weather west of the airport was moving closer with light rain and light turbulence starting to develop. Both of us immediately recognized we weren't climbing so I disconnected the autopilot; applied power and pulled back into the Flight Director. The speed was increasing so we retracted the flaps to 1. There was still barely a positive rate of climb at which point the FO recognized the speed brakes were still extended. The speedbrakes and flaps were subsequently retracted. ATC instructed us to level at 4000 feet. I leveled the aircraft at 4000 feet and we re-cruised the FMC to 4000 at a speed below 250 knots; then reengaged the autopilot and VNAV. It took a couple attempts to reengage the autopilot as I worked to trim the aircraft. The FO then began reprogramming the FMC for ILS Runway XYL. As we came around to the south we lost sight of the airport by a wall of clouds. I could see ZZZ2; which I initially thought was ZZZ after making the turn; before realizing ZZZ was completely obscured. We requested weather conditions at the field after observing various convective radar returns in the area. While I don't recall the verbatim response; light rain was reported by the last arrival on the opposite runaway; but we were told we were the first to land from the south. ATC then gave us vectors to the final approach course. Once intercepting the final approach course the latest radar return indicated a convective cell was directly over the field. I made the decision to go-around at that point as we could not see the airport and were not receiving realtime field conditions. As I executed the go-around we entered IMC and the ride conditions deteriorated rapidly. I observed lightening to the left and front of the aircraft.By that time we were in heavy rain and moderate turbulence conditions. I immediately initiated a right turn away from the weather while simultaneously instructing the FO to advise ATC we required an immediate right turn. ATC gave us latitude to turn at our discretion and assisted with a heading away from the weather. We were in a gear-up; flaps 10 configuration at that time. While in the right turn a gust of wind lifted the left wing causing a bank angle of approximately 35 degrees; which gave us a brief bank angle alert. I corrected the bank angle without issue. As we retracted the flaps to 5 and then 1; the speed increased quickly and over-sped the flaps by approximately 15 knots. The airspeed was stabilized and we leveled off at 3000 feet as instructed on an assigned heading. Upon exiting the weather; ATC asked our intentions. We requested a hold south of the field. ATC instructed us to hold as published at ZZZZZ1; which is the Runway XYL IAF. The autopilot was reengaged; the FO reprogrammed the FMC for the hold; we established ourselves in the hold and then took a couple moments to regroup. After two turns in the hold; the weather began to clear the area. Winds were reported to be calm and we could see the airport. We then successfully completed a second approach and landing to runway XYL. Upon landing we documented the flap overspeed and contacted maintenance for a required inspection prior to our return flight. No abnormalities were noted. In debriefing this event it was clear to me that I became task saturated with pilot flying duties and command decision making responsibilities. Trying to maintain a big picture without tunnel vision in a dynamic environment while facing deteriorating weather and ride conditions became challenging. Automation management became increasingly difficult as we were monitoring the weather situation; complying with ATC instructions and continuously reprogramming the FMC. Ultimately we had four approach set-ups to three different runaways; a discontinued approach; a go-around and a hold. I credit teamwork; communication and the FO's stellar pilot monitoring skills to maintaining safety during this event. A factor that negatively contributed to this event was fatigue. While we were fit for duty; this was the second leg on day 4 of 4 of the trip. The day prior was a long duty day extended by a maintenance delay and followed by minimum rest. There was also fatigue associated with the poor ride conditions and increased workload the entire way down from ZZZ1. The combination of factors certainly contributed to delayed mental processing while executing procedures not routinely conducted. During the debrief we discussed that using LVL CHG with an appropriate speed rather than VS would have been a better option. Ensuring and verifying the speed brakes were fully retracted prior to discontinuing the first approach would have also been prudent. Also; being more deliberate on the go-around procedure; including speed management would have mitigated the risk of a flap overspeed. Finally; we discussed that if we had it to do all over again; following the first approach attempt we would have requested a hold and given ourselves more time to evaluate the most current weather conditions rather than feel pressured to beat the weather. We felt like the decision to discontinue both approaches was appropriate and noted the positive application of CRM/TEM (Threat and Error Management) in successfully completing the flight.