CRJ200 First Officer describes a long duty day of IMC flying that culminates in a missed approach at DCA. With no real prospect of weather improvement at DCA the crew elects to divert. The Dispatcher strongly suggests that the crew hold until minimum fuel or attempt another approach. The crew declines.

Narrative

Our day started with an early show with IMC weather prevailing on the east coast. With six legs to fly in bad weather we had a full day with minimal or no breaks; so we had both prepared by receiving a full nights rest and showed up ready for the day. We operated the first five legs pretty much on time with all flights being in IMC with a high work load. We were not fatigued; however I am sure due to the high workload from the first 11 hours of duty for the day we were most likely not as fresh as we were at the start of our day. Captain and I started this flight on time by reviewing the flight and determined that we would be operating the flight in IMC pretty much for the duration of the flight and would be expecting to shoot the LDA/DME 19 approach in DCA due to the prevailing weather in DCA. We prepared ourselves and the aircraft for what we expected to encounter on the flight. We departed with no issue and began our flight to DCA. While enroute to DCA there were no issues or anything abnormal. We checked in with Potomac Approach and were advised to expect the LDA/DME 19 into DCA; which we had expected and prepared the aircraft and reviewed the requirements for the approach as well as briefed all pertinent information. When we began the approach the weather on the field was reported as 1;100 FT overcast and 10 SM visibility; but we were advised the aircraft ahead of us went missed due to not breaking out. We continued the approach; however; we [were] both mentally prepared to go missed if need be. When we got to the approach minimums Captain called the field in sight; I looked up and saw the runway and made the leaving MDA callout and started towards the runway. Very shortly after we ended up IMC at which point we both called missed approach and executed the missed approach. We then were transferred back to Potomac and given vectors to try the approach again. While in the vectors to try the approach again Potomac advised us the reported ceiling on the field had become OVC005 and was only getting worse. We sent Dispatch and ACARS and advised them of the situation and asked if we could divert to ZZZ1 due to the proximity for our passengers as well as having a precision approach; we received no response. After a few more minutes of delaying vectors and no improvement we then sent Dispatch a second message querying them again if they agreed this was the best course of action. We received a message from Dispatch telling us to shoot the approach again even though the weather was below minimums for the non-precision approach. We sent them a message back advising them we were unable and were then sent a response to hold until our BINGO fuel then we would divert. With the weather in the area (including our alternate) deteriorating; our fuel around 3;200 LBS; numerous other aircraft ahead of us holding for DCA we decided the safe decision was to divert to ZZZ1 where we could take on more fuel and wait out the low ceilings. We advised Dispatch of our intentions via ACARS and asked Potomac Approach if we could proceed to ZZZ1. We obviously had an extremely high workload due to just going missed; bad weather; trying to prepare to go to ZZZ1 and coordinating our intentions with dispatch. While flying at 4;000 FT with a very high work load our Dispatcher then sent us another message telling us to shoot the approach again and that the ceiling was not controlling and not to divert to ZZZ1. We advised him we did not agree with his decision and were proceeding to ZZZ1. They then sent us another scathing message telling us we were to hold until bingo or shoot the approach. We all understand that an aircraft out of position is a headache for Dispatch and the company; but our primary concern and responsibility at all times is the safety of our passengers; and we felt taking the aircraft to ZZZ1 was the safest course of action. We told Dispatch we felt ZZZ1 was the best option and we were headed there; they then responded again fighting our decision. At this point we were nearing the terminal area for ZZZ1 and nearing the ILS approach we were about to execute into ZZZ1 and did not want to argue with Dispatch and distract ourselves from our primary responsibilities. We landed in ZZZ1 in bad weather with high winds but had the aircraft and passengers safe. Throughout the next hour other we observed numerous other aircraft landing in ZZZ1 after diverting from DCA due to the weather remaining below landing minimums. I am writing this report because I believe Captain and I did everything we should have done to ensure safety of our flight but received instructions NUMEROUS times from our Dispatcher to continue trying to get into DCA for what I feel to keep the aircraft in position for the next revenue flight. In times of high workload especially with bad weather and after a long duty day; I would hope and expect that a company Dispatcher would solely have our and our passenger's safety in mind; in this situation it did not appear to be the case.