We travel with our car seat in one of these:
It's survived well over many trips and fits the regular baggage dimensions, though at LHR we have often been asked to take it to the oversize belt.

There's plenty of room and spare weight in the bag with the seat packed to shove in other bulky items. You might therefore not need a separate bag of your own, depending on how much of your own stuff you have.

We travel with a simple, light belt-in seat though. A rotating or isofix one will be heavier/larger.