As I understand it, and I could be wrong, but the SR measures your historical driving safety over an extended period of time, and is used primarily to determine what car classes and race types you are eligible to drive. DR is used to measure driver skill and the system primarily uses this to sort drivers into splits so that similar skill levels are driving together as much as possible. This is similar to the way iRacing manages SR and IR (DR in LMU), though the math behind them isn't the same. The driver badges look at driver behavior and safety over a shorter window of only 10 races, and are used to sort drivers with similar DR into splits with more similar recent safety history. So for example, if 40 drivers have similar DR, but 10 of those drivers are Green badges, they end up in the same split before filling in the remainder of the field with the next best driver badge (blue/good) and that sorting continues through the badge colors until the field is full.
The problem with SR alone in both games, is a blind spot that exists in both games when judging driver safety using only a long horizon metric; using only long horizon metrics allows the player to game the safety system by using the SR rating like a currency to be spent. When the safety rating is high, a driver can choose to drive much more aggressively, carelessly or even maliciously with little impact or consequences over a race or even a few races. The badge system in LMU accounts for this by also comparing a short horizon metric to expose the aggressive/malicious/careless behavior and tries to cull those drivers out of the best splits. How well this works depends in part on there being enough drivers overall, and enough with higher badge levels to sort effectively into high skill, high safety splits. If there aren't enough registered players with a green badge, then it's not possible to prevent padding out the split with progressively worse badge level drivers.
The system isn't perfect, but it is much better at preventing unsafe drivers from being chronic offenders than the SR alone system used by iRacing. I think I have how the system functions at a high level correct, and the explanations make sense, and I am sure if I have any of the details wrong someone will help with a correction, but maybe this helps a little.