First of all check with your dealer there is a fuel pump flange recall, there are a number of recalls the latest being the rear wheel carrier flange, all of course free to you, your local BMW dealer will be able to tell you which ones have been done.
On the fuel pump flange recall the flange can crack and lead to fuel leaks and there is a band that they use to secure it, but if the cracks are deemed bad enough you will get a brand new fuel pump as I did, but be prepared like anything with BMW its not a cheap part if you have to buy one.
It could be the fuel pump controller and I believe this can be by-passed ?
This is how its done on an RT
Other things you can do is make sure the flashing brake failure light goes out when you hit 5 miles an hour, if it stays on ( solid red ) it could mean the ABS module needs fixing or replacing.
Get the bike on the centre stand and rotate the rear wheel if it appears rough it is a sign of final drive bearings on their way out, also rock the back wheel with you hands at 12 o'clock and half past then at quarter past and quarter to, if you feel any play again the bearing is worn and you will need a final drive rebuild.
Check for ESA shock leaks.
I also believe there is a recall for the switch gear on K1300GT,s...I may be wrong but my mate certainly just had his replaced ( the new ones are made by Nokia )
You may want to get hold of a GS911 ( diagnostic tool ) and see what faults come up and see if it something other than the fuel pump ?
But if it does come up with fault code 10443 fuse activated to electric pump do not clear the fault code and take it to a dealer.
These are a few of the well known faults that seem to rear their ugly head quite commonly , but in reality you have a super bike there and I am sure it s nothing too serious to get it running again.