If your suspension is 'down' (into the weeds?) then your system has a leak, maybe not a big leak but big enough!! My experience is mainly with Mercedes (Great for the first 4 years/after that some oily rags, pertol and a lighter works best!!) and once the system loses a set amount of pressure the compressor starts to run continuously, which is not what it is designed to do. Again, in my experience the 'clients' didn't realise that because they had their ICE systems turned up too loud. Sometimes running after the car is shut off, or usually on initial start up(a clients wife Cayenne has this issue at the moment) the system just needs a 'top up' every so often. These systems main achillies heel are the rubber components and seals that deteriorate over time(decades in your car?) through lack of use, excessive heat etc just the usual stuff cars experience. The compressor running a lot or contionuosly indicates that issue as the sytem is no longer sealed. This isn't related to your fuze blowing though, unless Land Rover installed a 'fail safe' to prevent compressor 'burn out'. Not heard of that before but possible. The fact that two fuzes have blown, when the pump is cool(?) dissuades me from that. The compressor can be re built (there are kits on fleabay) and its not a too difficult job(loads of youtube experts have posted this kind of stuff)
I would get in amongst the wiring and check for any 'chafed' or 'eaten through'/burnt wires....It's usually the reason fuzes 'blow. Also check all connectors, fuze connectors, relays etc for melting/plastic 'cooked' or even just a snapped wire that is 'going to ground'....The smell of burnt plastic is a good indicator...especially if there have been 'added components' as mentioned before...
"I LOVE THE SMELL OF BURNT PLASTIC IN THE MORNING..........IT SMELLS LIKE..........BURNT PLASTIC'