Check that monstrosity at the earliest possible moment, don't attempt doing security theater with it. You will be happier, and so will the folks in line behind you.

When the kid is a little bigger, get a stroller. It will fit under your seat or in the overhead, saving you from having to argue whether that kind of luggage check is "gate" or "valet."

At 3.5 months the kid is likely a bit small for it. Until then, strap the baby to your chest and don't even worry about strollers in the airport.