As others have said you would need to check with your site operator about the use of an external gas cylinder, and storage of same when your touring. Not all sites will have the same arrangements as they are independent companies.
Regards to using an external cylinder with a touring caravan, you may have problems with finding a high pressure pigtail ( the connection pipe with the correct couplings from the cylinder to the gas pressure regulator built into the caravan) that is long enough to reach and external cylinder.
But apart from that the gas should work correctly.
Others have alluded to the dangers of transporting gas cylinders effectively loose inside a car . That is definitely a NO! Just consider some facts. LPG cylinders for caravans are "Vapour Take Off cylinders" This means the gas valve on the cylinder must ALWAYS be at the top of the cylinder, to ensure it only allows gas vapour to be released. If a cylinder falls onto its side that would mean the valve would no only be vertically half way up the cylinder, and it could be immersed in the Liquefied gas content rather than the vapour.
In the event the valve is not fully closed or it has been damaged, instead of just leaking vapour, which is serious enough on its own, the prone cylinder would be leaking liquefied gas. In the process of filling an cylinder of LPG the pressure is sufficient to compress the gas to the point it condenses into its liquefied phase. That is roughly a compression ratio of 260 times. If you leak liquefied gas that liquid will instantly try to boil and convert is self into its vapour phase which means it expands by about 260 times. so a small liquid spill becomes a much bigger gaseous cloud. And in the process of expanding it draws the energy it needs to vapourise from the atmosphere around it. Is temperature will drop enough that if you get liquefied Propane on your skin it will cause a serious frost bight.
Then there is the risk of having a highly combustible fuel lose inside an enclosed space such as a car. Pure gas will not burn becasue it needs to mix with air to gain access to oxygen. Broadly speaking you need a ratio of gas to air in the range of 4 to 15% to support combustion. it therefore does not much of a spillage to produce a combustible mixture inside a car. Also bear in mind there are plenty of electrical sources of ignition within a car. It the mixture ratio optimum not only will the mixture burn it might have the flame speed to be considered an explosive with the associated force to physically blow out windows, not to mention the concussion injuries such an event might impact on anyone in the car.
There is a legal requirement for vapour take of cylinders to kept upright, and to be secured to prevent them falling or otherwise moved, especially when being transported.
When towing your caravan the cylinder(s) must be secured in the gas cylinder locker, and turned off at the gas bottle or disconnected.