I think during those points in the game were you cannot progress, just focus on getting DNA, don’t worry about trophies at that time.
I make sure to have a timed incubator all the time, shoot as much as I can, use the sanctuary items, and get my battle incubator when I have hit a ceiling.
You can try out other dinos during this time too since trophies are not the goal.
Once you have some dinos leveled up or a new one to try out, go for trophies again. I do like 2 or 3 rushes for trophies a month since only your high count matters, the rest of the time I don’t care if I win or lose. I do however try to make one of those times the end of the month so I can start the next month in a good position.