The carseat tote I have is a Prince Lionheart, but they are all about the same size and thickness. Yes, it is a bit bulky, but you can shove it in the overhead bin while on the plane if you have to.
What I always did with mine was to use it to cushion the stroller from damage: I made a simple sheath to cover the upholstered parts of the stroller and prevent it from unfolding when handled. It is a piece of bright rip-stop nylon with elastic along the sides, and velcro down the open ends. It just wraps right around the folded stroller to protect it. (Also, when you hook the straps on the satchel, the best way to arrange them is to attach them to the back rings, then pull them frontwards, putting them parallel to the zipper opening down the front. That keeps the seat from dangling too low on your back and bumping against your knees.)
We don't carry our seat anymore b/c DS is too big for using it on planes, but here's how I handled it. When we got to the gate, I opened out my stroller sleeve on the floor, took the carseat out of the shoulder satchel, flattened down the satchel on top of the stroller sleeve, then laid the folded stroller in the middle of the satchel/sleeve. I picked up the ends of the sleeve and velcroed them closed, letting the flattened satchel act as a pad on the axle side of the folded stroller. This method allows you to send the bulky stroller satchel into the hold for the flight itself.
For the logistics of getting to the gate, we use what we call the hand rule. Count the number of able-bodied grownups you have, and multiply by 2; that is how many hands you have. Then subtract one from that number for every child you have along who is too old to ride in the stroller, but too young to walk unaided. When you get the final number, you know the maximum # of bags you can carry/drag. If you are using a stroller, remember that 2 hands are needed to control it; so while the person pushing it CAN use a backpack or carry a shoulder bag, that person cannot pull a rolling suitcase. (The stroller itself is a "free item" in this system.) When DS was still in the stroller, our number was 4, so I pushed the stroller and wore the carseat backpack. DH wore the diaper-bag backpack and pulled two rolling bags. Now that DS walks, but needs someone to hold his hand, the number is down to 3, though he does manage to pull his own rolling toy bag with his other hand.
Definitely check your luggage, and if you don't own a rolling pullman suitcase, think about investing in one (but don't go TOO large; check the airline size regulations before you buy.) I found that the trick to airports w/ babies is to cut back on the number of bags, and you can do that by checking a couple of large ones rather than attempting to bypass the carousel. Use a SmartCarte to get everything from the carousel, or tip a skycap to handle the bags for you. Once on the landside of security, help is available; it's only on the airside that you have to be really careful. (Oh, BTW, take the hood and any accessories off your stroller and put them in your checked luggage; the stroller is less likely to sustain damage if it is stripped of parts that stick out.)
Good luck & happy flying!
