I've been through a similar process, but other way round, Italy to UK when I bought my HP2 Sport from BMW Roma and imported it to the UK (I've been living in London for the last 15 years).
I do agree with aspes.
Importing a bike to the UK (pre-brexit) was a pretty straightforward and cheap process.
Not 100% sure importing it into Italy is going to be as easy and cost effective.
Most Brits hate the DVLA, but they never dealt with the Italian Motorizzazione

Unless you are particularly attached to your GS, you might find it way easier to just sell it in the UK and buy another one in Italy. Even just selling a vehicle in Italy is a way more complex process.
Also, an UK-corrosion damaged bike would be impossible to sell in the future in Italy (they are not used to that type of corrosion we get here). Keep that in mind.
Just to answer a few questions:
You should be able to ride the bike up to 6 months (as an Italian resident) on its yellow plates. Depending on where you are insured... most UK insurers won't cover you for more than 90 days overseas. There is talks about shortening this to 3 months.
You can take the bike before becoming a resident.
I shipped a few bikes up and down, it is cheaper than riding them (but less fun) and it should cost between 350/450 quid.
You shouldn't pay VAT on a bike of that age and with so many kilometers. At least that was the case pre-brexit (I think the threshold was 3 years and/or 6k km).
One of the users here on the forum might have more information on the paperwork/cost for the full ordeal, as he does stuff like this for his job. I hope he spots the thread
Keep in mind the cost of a new LHD headlamp.