Using LPG bottles as air tanks.

No, the same total volume of air will be used so the running time will be the same.

But the number of start/stops will be reduced which is the whole purpose of this job. It's the starts/stops that wear these things out.

doesn't matter.
those units are designed to run continuously, which is why i'm a little surprised its running in stop/start mode.
not sure with that model but i know bigger ones usually have a maximum amount of stops per hour, as stopping and starting is quite hard on them.

Stop/starts are related to pressure. When the tank pressure reaches the preset, it stops. When the tank pressure falls to the preset, it starts. Bigger tanks = less starts/stops.

Why would you run a compressor continuously - where would the surplus air go?

typically the compressor has either an unload mechanism (afaik basically like an engines throttle) or variable speed drive.
the reason for constant running is because they do not handle constantly stopping and starting. those type of compressors are generally made for industry that uses a fairly constant supply of compressed air. ie theres always a machine on the production line thats using the air.
also they want to be kept hot. being an oil flooded type, the oil needs to remain hot to stop water building up in it. if it stops and cools down water from the compressed air forms in the lube oil and they wear out quicker.
not sure what you use the oxy for but i would have thought an oil-less would have been much preferred, especially if its breathable oxy.

Not breathable oxy, but is passed thru a drier/filter unit. There is no need for it to be running non-stop as the demands are for high volumes in bursts of up to 20 minutes. The current setup has the compressor cycling on and off every few minutes during those high demand periods but adding capacity will extend those cycle times.

the extra capacity will most certainly help.
but i would also check if the load/unload controls can be used with the start/stop.
that will reduce the compressors output so it takes longer to get to max pressure. you might be able to do the entire 20 minutes without it turning off at all.