Air carrier Captain reported inadvertently descending below Minimum Descent Altitude on an approach in IMC without required external visual references. Upon recognizing the deviation; the Captain conducted a missed approach.

Narrative

We departed ZZZ for HRL not requiring an alternate and the TAF forecast showed stable conditions throughout the day. This was the first leg of the four day trip and the FO (First Officer) informed me he had not flown for a month. After we got the ATIS weather at the field; which seemed like it took longer than normal to get; we set up for the RNAV 35L as advertised. ATIS F which I believe stated OVC019 and 10sm then FO said it was changing. After a long pause; ATIS G was 330/9 2 1/2mi -RA BR OVC009. Certainly not what was advertised when we left ZZZ but still more than sufficient for the RNAV approach. We got no message from Dispatch along the way indicating weather had changed. Approach Control gave us vectors for what appeared to be just outside of AVUDE so I had the FO sequence us with a line off of AVUDE. I believe we actually ended up just outside BOTVE and while I was contemplating how to correct this in the FMS so we would know when to descend we were rapidly approaching AVUDE and this left us 500 ft. higher than we should have been approaching AVUDE. I attempted to descend so as to catch the VPI and may have exceeded 1200 ft. per min momentarily. Callouts left something to be desired but neither of us expected this approach to go down to minimums or even close. Maybe the FO called out minimums but I don't recall hearing it. I did look up as we were approaching mins. sort of in disbelief that I had not heard runway in sight to attempt to see the runway. When I looked back down I immediately initiated the missed approach and I believe I may have gone below the MDA by approximately 50 ft. This was not intentional and neither of us could believe this was actually happening. The go around went well and ATC asked us to state our intentions. I asked the FO what the winds were but got no response. I could have queried ATC but I knew they were near the tailwind limit so I instructed the FO to request the ILS backcourse 35L. My thought was it would at least get us a bit lower. With workloads already high there wasn't time to discuss alternates with Dispatch. We got short vectors back in for the ILS BC. We had excellent ground contact; the airport environment in sight and good visibility but a low hanging cloud made it harder to see the runway until we were clear of it and we made an uneventful landing with just under 1800 lb of fuel. This runway has no approach lights which makes it even more difficult. Low on gas and low on options was not how I expected this flight to end up. It could have easily resulted in a fuel emergency if we had to go around again.I think the biggest takeaway here for me is that I should have reached out to Dispatch for an alternate plan as soon as we got the weather in HRL. I'm not sure if I just misheard the FO when he read it off or what but it just didn't click that there was potentially going to be an issue. Neither of us thought we would have an issue with the weather getting in. I should have requested the current winds from ATC and this may have allowed me to request the ILS 17R but with wet runway numbers I don' know if it was possible. Also it would be a huge benefit to have approach lights or at least some strobes on the end of 35L. This would help to easily identify it through thin cloud layers or haze. Brownsville and McAllen were closer than I thought they were but I did not have current weather at either location. Again quick thinking should have been to get ATC working on an alternate nearby airport with good weather while we did the second approach.