A little bit of an update post. The game is actually possible to beat with no jumps under these rules. (2 if spike jumps are illegal)
I explain this in a different comment, but basically wait until the strawberry in 1700 is at its lowest, fall into it, timing it such that you can dash up and collect it in the same frame. You will get another dash, so dash up again, spike jump, then its a success.
Edit: You can dash-jump instead of spike jumping