Captain reported a misapplied MEL and erroneous Dispatch release resulted in a fuel issue that required a diversion to refuel and continue to destination airport.

Narrative

We began the day with a XA20 report in ZZZ1 to operate Aircraft X to ZZZ. Other than a delay due to customers; there were no other issues. A few months ago; I had an MEL regarding the BSCU (Braking and Steering Control Unit) unit that at first glance said we could not be dispatched to an airport with a 5G NOTAM. Another pilot and I looked deeper; figured out the MEL would allow us to go; although it wasn't clearly written. Of course; we would not get dispatched an airplane that an MEL would not allow to go to that destination. This was my mindset when we looked over this MEL as well. The plane arrived at our departure time. I never got the AREQ I requested which I like to study before a flight. Once the plane arrived; we were rushed. After reading the MEL XX-XX-XXX; the FO (First Officer) and I discussed where both center fuel pumps were deferred. We were confused; because we knew we needed the center tank to get to ZZZ2 and since we had 37.1 in total fuel; we were missing something. The front of the Release had the MEL listed as well as 37.1 lbs of fuel. Again; my mindset was we would not get dispatched with an improper MEL. After all; no one would dispatch a plane that was capable. I figured I was missing something; and the FO and I discussed it further. We both agreed that the scavenger pumps must kick on once the engines are started and that it was the electric pumps that must be deferred. We confused the system of the 321 to the 320 which doesn't have scavenger pumps for the center tanks. We again discussed also that the plane would not have been dispatched to ZZZ2 if the MEL did not allow it. Once on board; we noticed that the mode selector for the center fuel tank was left in manual by the previous crew. We looked; but nowhere in the MEL does it say to select it to manual. The FO looked up in FCOM that the FLSCU (Fuel Level Sensing Control Unit) controlled the valves for fuel transfer to the wing tanks; by placing the MODE SEL to MAN it would force open the FLSCU valves allowing fuel transfer. This part of the FCOM does not specify between the 320 and the 321. This enforced our thoughts that the scavenger pumps would transfer once the engines were running. So; since the MEL said the fuel was unusable; we thought only while using the electric pumps; but not when the scavenger pumps were operating. Which we now know; the 320 doesn't have any. Maintenance was on board for another issue; and we asked for their thoughts. We had the Fuel page up showing the 10.0 lbs of fuel in the center tank. Maintenance read the MEL and looked at the Fuel page. We discussed the FCOM and FLSCU; and Maintenance said we could leave it in auto. Now here is what I believe to be the misconnect. Thinking about the MEL I asked; 'If we leave it in auto; would it transfer fuel in the air?' Maintenance said 'Yes.' I truly believe; in hindsight; that during our conversation; I was asking about the fuel pumps working regarding the current situation; and I think Maintenance was not connecting it to the MEL but was responding about the system in general. Because our conversation involved more than just the MEL; a miscommunication occurred. However; this is the point where I felt confident to continue as I had done my due diligence and gotten clarification from maintenance: with the erroneous release; the unspecific model in the FCOM; and the discussion with Maintenance led us to mistakenly think we had the fuel as planned. Then we noticed in the air that the fuel was not transferring. We immediately got in contact with Dispatch starting with a satellite call along with Maintenance. Maintenance now informed us there was no way for the fuel to come out of the center tanks. Of course; this was the only thing we were discussing. The Dispatcher and I discussed our option; which included a stop in ZZZ3. At the time; we had an EFOB of 15.3 minus the 10.0 in the center tank which gave us an actual EFOB of 5.3. This is typical when landing in ZZZ4. We discussed it and decided to continue to monitor and make a decision as we went along. We discussed it every hour with the Dispatcher and the fuel stayed steady at an EFOB of 15.1-15.3. We decided abeam ZZZ5 would be our final go-no go decision. When we reached abeam ZZZ5 we were still showing 15.0 and the weather was fine at ZZZ2. After a discussion with Dispatch; we felt safe continuing. As we started the arrival; ZZZ2 Center shut down the arrivals due to a line of weather further east of the field along the arrival. We asked for an immediate turn back to ZZZ3. Once in the turn; the EFOB said 14.2 at ZZZ3 and ZZZ6; but 15.2 at ZZZ5. We elected to go to the nearest suitable airport.Once on the ground; we were quickly serviced; had the new dispatch; fuel and ZZZ2 had lifted the ground stop. We were wrapping up when the Maintenance desk called to say we were illegal due to [duty policy]. With the additional leg; it would have put us over 9 hours. We taxied to a gate and deplaned; where they were accommodated on another flight.