M.R.
We travel overseas every year and our daughter has done the trip since she was six months old so I know where you're coming from.
I am very surprised that the airline said no to a car seat. It's fundamental safety for any young child in any kind of transportation. So did they explicitly say "It is not an FAA approved seat so you can't use it"? What reason did they give for no car seat? I note that you say it's not a U.S. airline. I would first pursue this issue with them before anything else; your child really does need to be in a car seat that's strapped IN the airline seat for safety -- all the time, unless he is up walking around with you. I would truly get in the airline's face about this. Honestly, I would never have flown an airline that said my child could not be in a car seat.
Here's why: I've been in turbulence that would have had my kid bouncing like a ball if she hadn't been in a car seat. The adult seat belts on airline seats are NOT adequate for a 14-month-old. Your child will be able to slip around the straps, and will figure out how to undo the belt entirely -- it's easy, you just pull up on one big, flat piece of metal and you're free. You must not hold your child on your lap either unless the airline has given you (or you have bought) a special seat belt attachment that straps the child TO you for flight. That's not a great option for an entire flight.
Same thing with having him sleep on a pillow on the floor. If you encounter turbulence he will not stay put. Items could fall on him from above. A kid in his own, assigned seat, in a car seat that's held in by the airline seat belt, is the way to go for your son's safety.
You will get posts saying "Oh, it's fine to hold him the whole trip" or "He's fine without a car seat in his own airline seat." Well, if you would not hold him in your lap in a car, or let him ride in a car without a car seat, then don't put him in an airline seat that way either. Besides the safety aspect, you won't be able to stop him wriggling and trying to get up and you will get to your destination exhausted and upset.
We fly a non-US carrier and have never been told not to use a car seat.
The airline's trying to placate you with the "you might get a bulkhead seat" statement. Do NOT depend on getting a bulkhead seat, ever. The airlines will tell you it's possible but we got one once in 14 years. Increasingly those seats go to business travelers who are airlines' frequent flyers so those adults get more leg room -- the airlines are more interested in keeping those flyers happy than in accommodating you with your young child. Do ask about bulkhead seating (if you happen to get it, you can let your son walk around in that extra space a few times but do not just let him hang out or sleep there). But do not depend on getting it.
Otherwise: Take extra everything --especially food and drink, because if you have a delay and have to sit in the plane for a few hours (which has happened to us), you must be ready to feed him and not run out! The airline won't just hand you food for him if you're delayed. TSA restrictions mean you can't haul all the liquid you'd want so plan ahead to buy things like juices or water bottles once you are past the TSA checkpoint (which means they'll be expensive but you can't be caught without them if you have to sit on the plane).
Take new things to do and play with that he has never seen before. New books, new little things he can manipulate with his hands, new small toys, something like the "crayon magic" Crayola stuff that lets kids color on special pages with special markers but they can't mark anything else with those special markers (so he can't mark up the tables or seats etc. even by accident). Do not worry about spoiling him -- you really need new, fresh things to distract him. Yes, he'll sleep some of the time but you'll find that he won't fall asleep as easily as you hope and at the times you hope, so having things for him to do helps!
Please do not give him Benadryl or other over the counter meds to help him sleep. Those meds now have a warning on them that the are not intended for that purpose. If he's never had them before, and ends up sensitive or even allergic to them, you do not want to find that out in the middle of an international flight. And some kids are not sedated at all by Benadryl etc. -- it made my daughter wide awake instead, and doctors will tell you these meds get some kids keyed up instead of sleepy. Not worth it.