How much does it usually cost to relocate a heat pump to another side of the house?

I've moved two units over the years and both times the refrigerant was recovered with a pump and stored, then the system was put back under vacuum and recharged once the new pipe run was in. You don't just unscrew the joints and let it hiss out, any decent installer won't do that because venting it is bad practice and in many places illegal. So the answer to your main worry is no, a longer run doesn't mean starting over with a new system, but it does mean more labour and materials than you're probly hoping.

What matters is wether the new run keeps you inside the manufacturer's maximum pipe length for your unit. For most domestic monoblocs and splits that's quite generous, often somewhere in the region of 15 to 30 metres, and installers can add a small extra charge of refrigerant per metre if needed. Check the manual or the data plate before anyone gives you a number, because if your new run busts that limit then yes, it turns into a bigger job with a diffrent unit.

On cost, expect a full day or two for two people rather than a half day. They're pulling the old pipes out, running new insulated line down the side, extending the electrical supply, building or bolting down a new base at the back wall, plus recovering, pressure testing and recharging. Access to your back wall is the thing that will inflate quotes if the neighbours' garden is the only route, because carrying tools and a compressor through someone else's property needs thier agreement and slows everything down. I'd guess a four figure sum rather than hundreds, but nothing like the eye watering numbers you sometimes see quoted for full system redesigns. Get an installer round for a proper look before trusting any phone estimate.

One more thing, since the move only has to happen because your neighbour wants a carport, it's completley reasonable to ask them to chip in. Plenty of people in your position get at least some of the cost covered, and it costs you nothing to ask before the quotes start flying.

Not sold on the idea that the neighbour will chip in though. If the unit has been sitting on their side of the line for years, some would argue that's your problem to sort, not theirs. We had a fence line arguement with the folks next door and after the surveyor got involved it turned out we were the ones in the wrong, which killed any goodwill fast. I'd get the boundary properly confirmed before asking for money, otherwise you might be quoting a legal position that doesn't exist and souring things before the digger even turns up.

We shifted ours when we built the extension a couple years back. The bit nobody warns you about is the new base. Ours ended up on concrete pads the installer wanted laid level and ready before they'd even quote the rest, and I mixed and poured those myself one weekend to keep the bill down. Worth doing if you're happy with a shovel and a bag mix. Also the electrics got done by a seprate spark, not the heat pump firm, wich saved a bit and kept the quotes easier to compare. Ask whoever comes round if they'll work that way.