DCL does not offer a discount on Hyatt rooms, but rather a flat-rate that does not vary based on availability, unlike the Hyatt website. If visiting at a popular period, the DCL rate will probably be cheaper but it could be cheaper to book through Hyatt if travelling at a low period. You cannot stack other offers (senior, AAA, etc.) on top of the DCL rate.
We've also found the Hyatt rate for the same night can vary significantly over time.
We typically book when Hyatt first releases rates for our date, so about a year out. We then check often between then and our cruise to see if the rate has dropped -- if it has we book the new rate and then cancel our existing reservation.
We have found that booking when they first release rates is almost always the cheapest rate; sometimes we are able to save maybe $60 or $80 for the night at a later time when they temporarily drop the rate, but that is about it. In general though, they just raise the rate - though it bounces around, so it isn't like it is a just a steady incremental raising the closer you get to the sail date.
But this is also where the differential between Disney's contracted rate with the hotel and the rate you can get by booking direct comes in. We also saw this with our recent Alaska cruise and our stay at the Fairmont Vancouver Airport: When we booked our hotel ~13 months out directly with the hotel, the rate we got was about $200 less per night than Disney's rate; but for anyone booking their hotel say 8ish months or less out from sailing, Disney's rate was $50 or more less than what you could book direct at that time. By check-in time, our rate was about 50% of rack rate, and Disney's was a good chunk less than rack rate; the discounts the hotel was running at the time would save you 10-15% off rack if you were booking last minute and were eligible (AAA/CAA) or booking more than one night. Our Alaska cruise was rebooked 5 times over 3 seasons so we have lots of data on the hotel rates ;-)
Biggest lesson learned is to book your hotels as soon as you know you need them; don't wait ! Book a non-deposit, cancelable rate, and then recheck the rates often to see if you can get a better deal. Also check Disney's rate, as it could be a better deal, but make sure you understand the restrictions it has.