I have a 635d on a 2009 and I only bought it 3 weeks ago. With a budget of 12k you will be able to find one that is near on pristine condition! I paid £7400 for mine with only 2 previous owners and full BMW service history! I even have the original receipt of sale for £63800 ten years ago.
This is not my first BMW however it is my first 6 series and like you wanted to know what to look for and know what I was looking at being backed with as much knowledge as I could so I wasn’t parting my cash for as you say a lemon.
So from my experience first of all service history is a must!! Common issues with these are swirl flaps, it’s about £150 to have them deleted. From the information I’ve gathered on them it was a poor decision on BMW’s half to ever use them. And having them removed makes no difference to the performance of the car at all.
Second thing is the DPF, now the rough life span of the DPF from factory is 120k “roughly” and if they do fail it’s a £1200 job from BMW. But if you have the history you may find a receipt of it already being replaced (mileage dependant)
Another thing to look for is blocked rain gutters. I have been told that if they block then the water runs between the door and the actual interior door card. (But this another thing hat I read up on with these cars) mine seems to be ok (fingers crossed)
Mine is the sports with every optional extra bar the rear reversing camera. But touch wood I haven’t had any issue with anything and all is working as it should. There is another forum called bimmerforum.co.uk and on there, there is a special 6 series buying guide for you to look up. But these cars as said above are pretty bulletproof, as long as it’s been well looked after.
i would suggest with the budget you have to write down the exact spec that you want and the colour that you want to. Don’t buy one just for the sake of wanting one. Get one that has everything you want, once you find the one pay a bmw specialist to go view the car with you along with his computer to plug it in and run through any fault codes. If you buy from a Trader all cars now come with minimum 3 months warranty. But if you want there is a company called click 4 warranty in my option they offer the best cover but you must go for the 5 star service to be completely covered. also I would recommend lowering your budget to around 9/10k and keep a couple back just if anything was to go fail in the future you won’t be taking more out your pocket as you would have it left from your original budget.
I hope my advice is helpful. And here’s a pic of mine. I absolutely love it, it’s an amazing car and so much car for the money you won’t regret it as long as you do your research. Good luck and keep us posted Rik
these are before and after shots of my 635 😉