Hello Christopher,
You should use the type of gas and at the specified pressure indicated by that appliances data plate.
Whilst the vast majority of LPG appliances were rated to work with both Butane and Propane at 28mBar and 37mBar respectfully, if the manufacture has not specified Butane, then there is a chance that it might not work properly on Butane.
One consideration may be gas take off rate (or how much gas it uses). Butane bottles have a much lower capacity to sustain a high rate of gas usage than propane, and smaller bottles cannot provide the same high rates of take off that larger bottles can.
A fairly detailed explanation follows:
A new bottle of LPG (Liquefied Petroleum Gas) does not contain simply gas. It also contains liquefied gas. This is either Butane or Propane which has been pressurised enough to cause the gas vapour to turn into its liquid form. The pressure in the bottle is directly related to the temperature of the bottle. At 20C Butane will be at about 2Bar and Propane will be at about 6Bar. This is one of the significant differences between the two fuels. With no gas being used, the bottle will be dormant.
As soon as you use some gas, you have reduced the pressure a bit, and as a result the liquefied gas starts to boil and vaporises some of the liquid to replace the gas used. Just as with boiling a pan of water you need to add heat to keep the liquid boiling, and in the case of LPG bottles it gets it from the latent heat of the bottle and its contents.
Consequently when you use LPG, the bottle actually gets colder. How quickly this happens is dependant on the Gas Take Off (GTO) rate and the volume of the liquefied gas remaining.
As the bottle gets colder the vapour pressure drops, and eventually it will reach the point where there is no pressure to push gas out of the bottle. Normally the bottle will collect the heat it needs from the atmosphere, but if the weather is cold and/or the gas take off rate is too high, it is quite easy to cause a butane bottle to stop producing gas.
This is one reason why gas bottle should not be insulated, and why gas bottle lockers are well ventilated to allow the best access to the atmosphere to raise the bottle temperature when in use.
Exactly the same process applies to Propane, but because of its much higher vapour pressure, it will continue to gas off at much lower temperatures than Butane. Butane becomes ineffective at delivering gas at about 4C so that is why it is not recommended for deep winter use. Propane stops delivering at about -40C so there is much greater potential either at low temperatures or for large take off rates.
Your BBQ, may try to use more gas than the normal butane bottles can supply, which is one possibility that the manufacture may only rate it for Propane usage only.
Just as pointer on this Gas Take Off rate and bottle cooling, the next time you see either a roofer re-felting a roof, or one of the road white lining contractors working, you will note that they use propane, and that frequently the bottles have started to frost up at the bottom where the remaining liquid is.
It is also the case that the max 'GTO' of a bottle drops as the liquid level decreases. This is because the surface area of the liquid in contact with the bottle walls diminishes, and thus the ambient heat collection area also diminishes.