B737 Captain reported a bank angle warning while diverting to an alternate airport due to thunderstorms and turbulence. The flight regained control and landed at the diversion alternate.

Narrative

While descending on the ZZZZZ X right near ZZZZZ1; we began to get vectors for spacing/ATC saturation and weather build ups to the northeast of the ZZZ airport. There was no sign of major disruptions until ATC began to tell aircraft that holding was to be expected at ZZZZZ1. However; I thought it was confusing because ZZZ ATC was trying to figure out what to do with the airport and aircraft which I believed caused some critical situations for aircraft. At that time; ZZZ had microburst alerts on all runways which had caused a complete stop of arrivals/departures at the airport. After hearing this; I messaged the dispatcher via ACARS about what was happening at ZZZ. He had told me standby. The First Officer (FO) made a PA to inform the passengers that we were receiving headings around the airport due to the rising weather concerns. Right around that time; we received holding instructions at ZZZZZ1. After computing BINGO fuel; the FO and I saw that we had no time left to hold. In fact; we were slightly below BINGO fuel. We knew we had to get out of there. Therefore; we promptly told ATC that we needed to leave the hold as we were past BINGO fuel. Our flight plan alternate was ZZZ1 at the time. However; I personally didn't like the idea and flight to ZZZ1 because there was a thunderstorm cell that I saw near ZZZ1 and I didn't want to fly there and be in a bad situation with that weather nearby. Therefore; I messaged the Dispatcher and told him I wanted to go to ZZZ2. Then; the FO and I changed the DEST on RTE page 2; then completed the DIVERT* prompt in ACARS. I sent one last message to the Dispatcher to finalize the decision to go to ZZZ2. After this; we began to receive vectors to ZZZ2. On the way to ZZZ2; the Dispatcher messaged us and told us to consider ZZZ3; but we were already south of ZZZ and the fuel burn to ZZZ3 was more than it was to go to ZZZ2. Right around the same time; I transferred controls to the FO because I needed to inform the flight attendants (FA's) first; then the passengers of the situation. Then; ATC told us there was a microburst alert for Runway XXL at ZZZ2. The FO and I looked at each other in disbelief. We considered all options at this point but we were time and fuel critical. We queried ATC and asked how long does a microburst alert last for. Shortly after that; ATC said that the microburst alert had dissipated and there were only LLWS advisories. The FO and I were relieved for a moment. After I did that; the FO and I's next task to begin to set up; brief; and prepare for the approach into ZZZ2. We were going through turbulence because we were low altitude (somewhere between 11;000-12;000 MSL) and going slow. The FO disconnected the Autopilot to correct us back on course after turbulence shook up off a heading. We received a 'BANK ANGLE' alert; however; the FO quickly corrected while I had my hands on the controls for extra help. Furthermore; we completed the briefing and preparations for the approach but we were still getting vectors near ZZZ2 to wait our turn in line for arrival. We had alerted ZZZ1 Approach that we couldn't wait too much longer as we were burning fuel and getting closer and closer to minimum fuel which could result in an emergency fuel situation. We did not declare any emergency; but more so advised ATC situation. We received landing clearance and made a safe and successful landing on XXL without any issue. Then we taxied to the pad to the west of the airport to begin diversion recovery procedures. However; a storm cell to the west of ZZZ2 had developed rapidly and moved over the airfield. The ramp was closed for around 3 hours and we exceeded flight duty period. The FO and I followed FOM guidance for flight duty period. Eventually; I coordinated with Dispatch and ZZZ2 to receive a gate. There passengers were mainly compliant but I know sitting on an airplane at a ramp for 3 hours isn't ideal. They received water; snack; and access to the lavatory for the entire time. After taxing to the gate; we offered egress at the gate; I called the Dispatcher and coordinated a return to ZZZ. After boarding the airplane up; we left ZZZ2 and made a successful approach and landing in ZZZ to end the night.