A bit of history on this bike. It was owned by my brother in law and we had gone on a couple of tours, he on this Honda, and I on my same vintage BMW 900. I could get 190 miles on my smaller gas tank, while his range before hitting reserve was about 130 miles. So this bike has been a consistent gas hog. It has been sitting in his garage for the last 10 years and with no time to ride, he sold the bike to me last fall.
The K5 and R90/6 had the same size fuel tank, (18 litres) unless your BMW had the Euro spec 22 litre tank, which all the Aussie BM's had. BMW's were easier on fuel though, with only 2 cylinders and 2 carbs, so that's no surprise either.
The best fuel economy I got from a big bike was from my 1985 Harley Sportster with only 1 carb, but that was a good thing, considering the size of a Sporty's fuel tank.
Losing weight and getting fit is over stated. In the old days the only fat people were kings, and they lived much longer than the skinny-arsed peasants who made up their kingdoms. Don't sacrifice your "Royal" proportions for better fuel economy mate, some people are just jealous because they have no royal blood and can't grow a decent gut.
If you do decide to get fit, do it slowly, my ex-brother in law was a cop, and he told me once that "The Morgue is full of track suits", which was a reference to the guys who've had a heart attack and died while trying to get fit again. I think dying while trying to lose weight and get fit would really piss me off! (Oops, wrong thread..........)