Thankfully the days of brining bulky car seats are behind me , but as others have said there are bags that you can get for them .. some of the manufacturers produce them as "accessories" or there are generic ones you can easily get on Amazon/eBay .. we had a manufacturer's one and it worked fine for a few years , though eventually got destroyed .. but it served its purpose and went all over the world .. as others also correctly note it's a great way to increase nappy carrying space too as there's a lot of 'wasted space' inside the bag you can easily fill.

The car hire company ones are okay generally .. they mightn't be super clean sometimes .. but my main issue with them always was the cost .. on more than one occasion the cost to rent the car seat would have exceeded the cost of renting the car which is obviously nuts !

Unlikely they'll lose your car seat in transit, but if they did you should be able to claim the cost of renting one from BA or your travel insurance .. or perhaps better still take note of where you'd be able to buy one on arrival and just go buy one and claim that ( there's plenty of big box 'retail park' type shops very near FAO airport where if you're arriving during the day you'd easily be able to buy another one )