I guess you should start accepting a "nightmare" scenario: I doubt that Easyjet will check your luggage through to DXB. You're talking about an LCA on one hand and its complete opposite one the other, so don't expect any good will gesture from Easyjet. Moreover, there is no such thing as an Emirates transit desk at FCO, but even if there were, they would never send an agent to pick-up your luggage and check it in (they would not be allowed, anyway).

In other words, what you foresaw in your message is the bitter reality. Given your 1 year old, a potential delay (even a small one) combined to a potential bussing, a wait of 30 minutes (good average) for your luggage and so on, I would strongly recommend an earlier flight into FCO. 140 minutes may turn out to be fine but it is actually overoptimistic. And given the different tickets, you risk a cancellation of the onward one...

PS By the way, welcome to FT!