Since you can't get the Suzuki DR650 in the UK ... perhaps consider either the Yamaha XT660R (not the Tenere') or the air cooled XT600E. The 660R is 165 kgs. dry, (363 lbs.) the XT600E: 156 kgs. dry (343 lbs.).
The XT660 Tenere' is 183 kgs. dry. (402 lbs.) Once all are fueled up and measured for true Wet Weights ... all are heavy ... but the 660 Tenere' bumps up over the 200 kgs. mark. Not ideal for a travel bike intending off road use. Sure, it can (and has) been done. But is it fun or easy?
Both the other Yamaha's are substantially lighter weight than the Tenere'. I'm sure there are still plenty of XT600E's around since they were discontinued in around 2004?
The XT660R is simply a pared down version of the Tenere'. Lighter, simpler and the price is quite good too for used ones. (cheap)
Of course ALL would need plenty of up grades & substantial modifications ... as would nearly any bike you buy ... Unless you buy a freshly re-done travelers bike that is already set up to go. Something like one of Ted's bikes ... all the thinking and good modifications have already been done.
If you ever choose to buy here in the USA then you've got more options.
The Suzuki DR650 is just 324 lbs. (Dry) about 367 lbs. (Wet) Many for sale with lots of extras already done ... and cheap as chips compared to UK pricing. Also, the versatile and inexpensive (and hugely popular) KLR650 is also an option you don't have in the UK. All good bikes.
If you want to continue with BMW ... then, IMHO, the real choice is to take a virginal X Challenge and do as many of the Colebatch modifications you can afford or have time for. Once fettled, it will become a throughly competent RTW weapon. But think THOUSANDS of pounds rather than Hundreds to make an X Challenge right.