The only reasons I sold my easy to maintain and otherwise bullet-proof rocket ship 1987 Honda CBR1000 F-H in 2012 was there was only one new fork stanchion left on earth and the Dutchdealer was asking €800 even then. That and the rubber carb diaphragms were leaking. They are part of the slider mechanism and can't be replaced. There was a place in the US that remade them for $400. For each slider. That's $1,600 before shipping and taxes.

Running an older modern bike, rather than a classic, is a dodgy affair. You need an injected bike where everything can be rebuilt.

There are so many excellent 20 year old bikes out there that buying new for most of us over 40 is questionable. Those top end sportsbikes you lusted after but couldnt afford in your 20s? For example a good 2005 GSXR 1000 reliably poking out 150 to 160bhp can be had for £5k and goes faster and handles better than most people can handle and is cheaper to maintain . Why would you fork out 15 to 20k for a new bike if the pennies have to be counted?  Even on finance.

Sensed their was something by the tone of your post and why the dealer took the easy way out cuz my gut says tooth off the starter wring which is going to be a major ball ache compared to putting a new starter in,although while looking at the easy options I’d also go lithium battery if you haven’t already.

This discussion on k-bikes is a little scare - so i say book the bugger in soon and use the big K!

EDIT 2: This discussion seems more positive and a happy ending!

Good searching, Matt. Thanks.

The first link is for a K1200RS, but according to this, the K1300S has a 'starter sprag gear':
https://www.k-bikes.com/threads/rebuilding-k1300s-engine-easier-than-youd-think.51708/

From the K1200RS link:
'Many metal fragments from the sprag were throughout the crankcase.'
Since the bike was also serviced and no mention if metal chunks came up, I thint I can tentatively discount that one. With fingers firmly crossed.

The second link is also a K1200RS, but that was solved by cleaning. That would be sweet. I will read further.

The starter is spinning but the teeth on the  starter are not engaging with the flywheel. Show me a picture of the starter from Real OEM whatever (cos I'm idle).

I can't find a suitable picture of the engine in RealOEM.com. This is the starter motor - probably not useful:
https://www.realoem.com/bmw/enUS/showparts?id=0508-EUR-09-2015-K40-BMW-K_1300_S_0508,0509_&diagId=12_1358

And this YouTube vid shows the position of the starter motor (between the tank and engine), and it's end:
https://youtu.be/Mq1uwLpJzE4?t=309

Can I digress? Sounds like you have the bike on some kind of BM maintenance thingy or is that me just reading wrong.

Correct - a 5 year/30k plan which was a proper bargain at £210 per service, inc the big ones.
But I cancelled the warranty unfortunately, in a fit of pique at Warwick Shytner doing such a sub-standard bodge job last year. Some you win, some you lose! Hey ho.

I have an intermittent problem with starting.

Listen to/watch the video:
www.poqit.com/starter_fail.mp4

It only does this once in a while - maybe every 5th / 10th time. Occasionally (as in the video case) it doesn't start several times in a row.

I took it into BMW to investigate, and they couldn't reproduce it.
I then took it out several times until it failed again and then took it back to BMW. They listend to it (it only failed for about 2 seconds on the BMEW NOG premises and then fired, but that convinced them to replace the starter motor, just before Christmas.

It's taken a while to get back out and ride due to weather, but now I find it's not fixed.

I know the first step is to take it back to BMW, but does anyone have any hints on what else might go wrong - since it's not the motor itself?
(Actually, the ideal first step is to break it permanently so it can be diagnosed properly, but whatever.)

Thanks in advance...

Can I digress? Sounds like you have the bike on some kind of BM maintenance thingy or is that me just reading wrong.