Tiffins is a dining experience that can easily take more then 1.5 hours. It's not to be rushed and they don't rush you, they make sure you have the time to enjoy your dining experience. It's well worth dining there.
Also, very few table service restaurants seat you at the scheduled reservation time - they tend to run late.
I'm not sure how many are in your party, but there seems to be availability prior to 5:50 almost every day at the end of January for parties of various sizes. Using MDE to check is just giving times grouped close to the time selected, so if you try 4:00 and then 5:00 or the half hour options, then you get different results despite MDE saying it's searching a time frame that includes those times. Also, call the Disney reservation number.