We have done both. It is not like WDW where you have to avoid the EE park because it is more crowded--as DLR is a hoppers paradise--- BUT it is better if you are going at rope drop to go to the park that has not already been open for an hour. EE is great at DLR-- actually DLR is typically great the first few hours of ANY morning-- the lines typically build much slower than at WDW which I think is because of the local crowd. Our problem with EE is that the times we are there it typically starts at 7:00 a.m. Park open at 8:00 a.m. with our time difference is easy-- but even with the time difference 7:00 a.m. is a bit rough. We have done it for RSR when it was new and we did it to make the A&E dash when they were hugely popular with long lines-- but other than that, we typically just stick to park opening.
I will say that it sounds like you do not have hoppers? Be aware that hopping at DLR is nothing like hopping at WDW. To a WDW veteran, DLR is just like one big park. They are separated by a small plaza with bag check on the outskirts so there is no bag check to hop. It is 5 minutes to get out of one park and into the other. We have literally been from the back of DL to the back of CA in 10 minutes. We hop all day long. For example, if the lines get long in the afternoon at DL- we check wait times and then hop over to DCA when the lines are shorter. Last year we hopped over mid-afternoon and wound up riding RSR single rider walk on about 3 or 4 times in a row in the middle of a July afternoon. Also CA often closes before DL- so we often leave CA and then get in some extra time at DL. Last we saw WOC at CA, and then hopped to DLR and saw the late parade of Paint the Night- and we found a great spot for it just about 15 or 20 minutes before the parade. It was terrific because our PTN stake out night was planned for the next night, and we got all that time back that we would have been sitting on the curb the next night. (Actually- since we had seen it- we did not stake out a spot for the early parade the next night for a couple hours- BUT we did stay at DL and catch the late parade again with a minimum of stake-out (15 minutes) because it was SOOO GOOD!!!!!).