Has been very cold last few days. Is the bike kept outside, or in a shed?
Your voltage figures at rest and when running sound good. But you should put a voltmeter across the battery while cranking (cold) and see what it's like then. Let us know the result.
Apart from cables, have you checked that big fuse in the starter solenoid for damp/corrosion, and vaselined it? Same for that big earth nut (new cable there also) on top of the gearbox?
At your mileage, it may be worth giving the starter motor itself a service. Remove, clean up the brushes and everything around the commutator, check spinning cleanly in bearings, etc. No-one recommends it, but I tend to put (just a few) drops of WD40 on the bearing surfaces before reassembly. Sometimes new brushes and general cleanup can help, as it sounds like your bike is very often starting in a damp environment.
Any other vehicle, we shouldn't have to worry about all this. But since the battery and starter motor combo are only just powerful enough for the Tiger, we have to get everything near perfect.