@PeterL As mentioned already, I would not even consider a 320i in the E90 shape. In the F30 shape it would need to be the LCI model with the newer B48 engine as the F30 320i with N20 is also very problematic.
E90 320d is a lovely car, although infotainment is outdated. Good luck trying to find one in good condition at a good price though.
The links you posted, the first one has some changes to the exhaust and no xenon lights. I feel it is a bit pricey for what it is.
As you have just seen, it is not difficult to change the mileage on BMW's, more so than some other cars but it is achievable and it is done on a fairly regular basis by the people who buy them written off and fix them up to resell.
Generally speaking these cars do not have the spare key, also the service history is often missing or they only have up until the motorplan ended. Otherwise there is generally a gap in terms of time on the service ie. The 145k KM service was done in 2019 and now it's 2022 and being sold with 153k KM on the clocks - does not add up that the car would have barely been driven for that long and there should have still at least been the yearly service done. This is the BS the people like to try and pull on you when mileage has been tampered with.
For accident damage, check what is easy - do the body panels and lines of the vehicle line up (top of the doors, spaces between doors and all body panels, bumpers, bonnet etc.), check the stickers on the lights for date of manufacture, do they match and are they inline with the year of the vehicle (2009 model with lights from 2010 or later means its been replaced).
Also take a look and see if there are any mods or things which do not add up - like Msport bumpers but normal steering wheel, or Msport rims with nothing else Msport, rims from F30 or a different range (often they are replica rims and where are the originals?).
Small things like these can help you avoid a bit of a lemon as I feel these cars often have had a hard life or some dodgy shit has gone down with them.
With that being said, you can't go wrong with an E90 320d if you can find one in good condition. They have decent power, they're reliable and decent with fuel consumption and maintenance is lower than other cars and probably cheaper to maintain than a Honda, Volvo or newer VW.
