I have a meter and can test your battery and alternator. I also have a trickle charger (optimate) if you want to leave the battery overnight.
If you can pop over by me (I live in Peasedown St John), I can test the battery and charging system and give you some reassurance.
My hayabusa has a battery that started the bike on the first try. However if the bike did not start on the first try, the battery then acted as if it was dead. I removed the battery from the bike and put it on the charger. Now the bike and battery work perfectly.
With respect to your specific situation, I would say your fuse is good as the bike started and ran when jumped from the car. Also the bike battery got a quick boost from the car (but not enough to charge it fully).
However at idle the bike did not charge the battery and so used up any remaining charge hence the cut off 20 minutes later.
