Any of the above would work, but I like to take my own.
option 1- to rent one costs more than buying a new one - last time I checked it was £7 a day * 14 nights = too much for a car seat of dubious quality.
option 2 is fine, you can probably get a carseat quite reasonably, but what will you use between the airport and the store? Personally, I would be too stressed = also will you want to go directly to a store to find one when you are tired and really want to be finding your hotel and going to sleep.
option 3 - no hassle, just check the car seat and collect it with your suitcases, you know exactly what you are getting and your baby will be familiar with it. Also, you know exactly how to fit it, which is a big safety consideration. Also, if you pay for a seat on the aircraft (usually once baby is 2), then you can sometimes use the carseat on board the aircraft - we had mixed results with this, Airtours wouldn't let us on the flight out, but encouraged us to use it on the flight back. To be honest, my dd was a bit older (3) and she objected to being in the car seat when she knew she hadn't had to use it on the way out. It ended up being a bit of a hassle as there was nowhere to put it, so now we just check the car seat.