At 3.5 months an umbrella stroller would not really work.

If the little one has their own seat (not lap infant), I would baggage check the car base (or leave it at home, I do believe most of these car seats can be properly fastened with a seat belt without the car base), gate check the stroller (wheeled base part) and take the car seat on the plane. These types of infant car seats should fit, but I've had issues on low-cost airlines with smaller seat pitch, so beware. If it's a lap baby, then gate check the seat as well. As someone said, you will need that stroller for getting from one gate to the other, etc. As the kid grows, then a baggage-checked car seat and gate-checked umbrella stroller would be the way to go.

One caveat. Some airports, or certain gates at these airports, do not deliver the gate-checked items to the jet bridge. I have terrible memories from LGW. Our 7-month old did great in her car seat on the flight, but then it turned out that the gate-checked stroller part was going to wait for us at baggage claim. It was a good 20-minute walk from the gate and no fun lugging that baby in the car seat along with our other carry-ons.

If you have checked luggage allowance (or even if you had to pay a small sum), it makes a lot more sense to baggage check as much as possible when travelling with an infant. Only take the stroller, a backpack and/or laptop bag and of course the diaper bag.

Edit: I just looked more closely at your stroller. It looks like the stroller seat is affixed to the wheeled base, which might affect the relevance to you of some of what I wrote. Ours was the type below and my advice about gate-checking the car seat was based on having to use it as the stroller seat: