I afraid I don't think the gas and the light are connected.
Re gassing is quite common on all cars: Worth doing but take a look at all the pipes before spending money. You can often see a blue/green stain from the lubricant indicating a leak at a union or a fractured pipe, which is well worth fixing before wasting money on tests/gas. The compressor is easy to spot and should be pretty clear if its cutting in and out.
The light should come on when the system is active, even if it can't push out any cold air due to the absence of gas. Could be a duff sensor or a duff switch. The sensor is behind the radio panel and it is worth carfeully blowing the fluff out of it but you may end up replacing.
GTV and Racing Driver