The main deterant (for me) would be that the battery would have no regulation in the event of a short. Ok technically it would due to the back emf produced by the batteries internal resistance, but that is not going to be issue enough to stop your wires, controllers, etc from getting toasted.
Using a sine or square wave inverter and subsequently re converting to DC would likely consume 10-20% (or more) of the power of the system due to loses in the inverter/rectifier. May not be an issue really since your thinking of a mobile show, but that would depend on the output current needed and the size of the alternator on the vehicle and any other demands on the alternator. I would say RJ is correct. A DC/DC converter would be the best choice.
Since we are only talking ideas, you could use a good compact AC generator. I would suggest a field load though on the generator since their regulators typically do a better job regulating 2500 to 2900 watts than they do 0 - 400 watts.
On a side note, if you're dead set on using batteries, you could make some Edison cells (DIY style!) and string 10 of them together for 12v. Charge density is a bit of an issue along with charge efficiency, but they practically never wear out! (sorta expensive to make right now though and probably not very compact.) Also if you did end up with a 13.2v battery, since that is only a 10 percent variance (12.6v is only 5%) that would most likely be within the tolerance of the 12v components, but someone with the tech sheets for all the components would have to make that call. Ironically that article says car batteries are not 12v, but spends most of its time talking about the charging requirements of different composition batteries instead of the actual battery voltages. Good read though!
