Air carrier pilot reported making a fatigue call after experiencing multiple delays; crew scheduling issues; and not being able to meet the rest requirements of FAR 117.

Narrative

I am happy to provide great detail for a fatigue call I felt cornered to make even though yes; I was extremely tired; I also should not have been considered legal to fly. Prior to this trip I received normal rest and felt well fit to perform my duties. Date was the start of my trip beginning with a 5 leg day and after our first flight to ZZZ; we were stuck on the ground for 3:XX hours due to a large cell that grid locked State from ZZZ1 down. I alerted Dispatch; ATC; and Crew Scheduling of this situation and suggested that scheduling assign our ZZZ2 turn to a reserve crew to keep all flights on time and avoid a rolling delay and the possibility of our crew timing out; but this was ignored and so all flights aside from our first flight were a rolling delay. After additional delays due to weather and maintenance; our crew was then required to take a 2 hour extension; leading to 7:XX of blocked flight time; 13:XX hours of duty and 12:XX hour layover. The day of my fatigue call began with rolling delays without even enough time in between to stop and eat.Our crew did our best to make up flight time and turn time to get the delays minimized; however due to a heavy snowstorm in ZZZ3; we had to wait in line to be de-iced. We ended up over blocking over 1:XX hours on our flight back to ZZZ4. If scheduling would have built out the trips correctly showing a 40 minute turn in ZZZ4 and a 3 hour blocked flight to ZZZ5; it would have showed us going beyond our 13 hour FDP (Flight Duty Period) for Date 1. The trip had been manipulated to show a 3X minute turn and 10 minutes less of block which with 15 minutes of post duty time; we would have been still past 13 hours of block. Since I had been extended the night prior; I then need 30 hours of rest before being extended again. When contacting the scheduler about this I was told that the situation was being monitored and that updated number were 'in the cue'. I was told to hold off until receiving a push notification and a call from scheduling. At this point I was definitely fatigued and was going to generate a report regardless; however I did not want to have to use the fatigue call and deal with the negative ramifications of doing so (i.e.; sick bank reduction and reduction of min guarantee/tier pay) considering that this was a legal issue as well.After receiving the updated data that clearly showed that pushing back 5 minutes prior to receiving the 'updated' numbers that we were going to arrive at ZZZ5 beyond our FDP; I then called scheduling again to be released legally. We ended up conferencing Chief Pilot to discuss legality. I was told that if I push right now; 10 minute taxi to take off and 2:XX of flight and 5 minutes taxi into ZZZ5 that I'd be able to make it with the 30 minute extension. All of that was predicated on being fully boarded and pushing back at that moment (XA:45pm) and taking off no later than XB:05pm with only 2:XX flight time which we were not at all in the position to do. The math was completely skewed as we hadn't boarded because we were told to hold off since we were very close to illegally crossing into our hard 117 limitation. One minute beyond would result in a violation of 117 rest requirements and is completely illegal. The Chief Pilot was asked how to proceed by the scheduler and stated that if scheduling said we are able to make it happen based on their timeline then we should try for it. I expressed the unrealistic ability to make the flight happen and the situation of violating a 117 regulation and still was made out to feel that I was the cause of our delay. If anything happened in flight as in ATC slowing us down; giving us a hold; deviating around weather (which we very well may have needed to do) or simply not had a ground crew ready to taxi us in; I would have been in the situation of needing to divert; open the flight deck door prior to block in or declare an emergency to get to an airport in time before violating my 117 rest requirements. I knew my personal limits as to my own health and well-being and was certainly tired regardless; therefore was forced to use the fatigue call even though using realistic numbers; we were not legal to fly but was; in my opinion; being bullied into trying to make the flight happen at all cost and that to me is a big safety concern.I believe that crew scheduling needs to not pressure pilots into potentially violating 117 regulations. It is a joint responsibility of our team to ensure that the flight is safe and legal to fly. They also need to do a better job of collaborating with the pilot group. This entire situation could have been avoided had scheduling removed the ZZZ2 turn from our crew the night before as I had suggested. All flights aside from ZZZ-ZZZ4 would have been operated on time. Instead this lead to out crew worked to the extent of our legal limits during fatiguing situations such as weather; maintenance and delays. I understand that we are in a tight situation with Pilot availability; however it is most important that flights are being conducted safely and in correspondence with 117 FDP limitations.I believe the root cause might be the culture that we are in currently. We have a very short staff of reserve pilots at the moment and so I feel that mostly we try to stretch our pilots thin in order to satisfy the consumer demands of travel. I would have loved to see that flight go and get the customers to their destination; however it is my responsibility to operate that aircraft safely and legally. It should not be the culture to pressure pilots into making a flight happen that will potentially cause a violation. This culture is a safety concern and must be changed.