Hi Mark and welcome to the madhouse that is EuroKClub.
There are a few members in here who have owned both the K12S and the K13S, black-k1 (David), TomL (Tom) Phmode (Brian, myself) and a couple of others.
All agree on one thing, if we were in the market for a K today, even with a limited budget, we would go for the K13 over the K12.
The K12 had niggling fuelling issues, dodgy (but VERY powerful) servo brakes and a rather wooden feeling chassis. I'm not saying the 13 is perfect (hot start issues, paint peeling, clutch squeals and judders etc.) but I would NEVER buy a 12 again.
With 5 owners, you have to ask yourself 'why?'. Mine had four previous owners and I found out why the hard way. None of them could cope with, nor fix, the fuelling issues.
Only part service history with a K is a no-no! Privately serviced (like TomL, TomK1300S and many others) complete with a wad of receipts and entries in the service book are OK too. Patchy service history? Nope, sorry.
These bikes can be fragile and finicky and my advice would be:
Don't buy a K12S, spend a little extra on a K13S, you will (probably) spend the extra money in the end anyway if you buy an early K12S, and possibly a later one too.
If it has to be a K12S, then buy at least a 2007 onward model with at least a full dealer service history (correct oils and brake fluids used and on time), the fewer owners the better (bike loved and cared for, not moved on quickly when something goes wrong) and take out the BMW Extended Warranty.
Shop around, get a well maintained, low-mileage K13S and you will be far happier in the long run.