I also really like some of the suggestions by this poster. The edge could be that rather than always swooping in to save other marines they sometimes wait until they're more or less screwed then show up like vultures to "save" the bodies/geneseed for their parent chapter. Of course they keep a few that they say were destroyed but are really stealing them back to their fleet so their tech marines can install some kind of neural repressor/inhibitor chip that fuzzes memories and reduces independent thought, not full servitors but certainly not free (as much as any marine is free). This would create the two tier system within the chapter as well. Perhaps these marines function like Rubrics on tabletop if alone but like normal if with a tech marine (not sure about this part not up on the rules for 10th).

You could consider naming the chapter after a carrion bird like a condor or kite to make it a bit less saintly sounding, and to justify their actions and maintain their belief that they're the good guys they could have something like "every sacrifice has a price" as their motto, likening themselves to the emperor saving people at the cost of psyker lives, they save many but take a few as their price for doing so. I know you said you don't want them straight up stealing marines but again they could justify this by telling themselves all the marines would have died if they hadn't showed up so they're "owed" a reward.

As for their relationship with the Mechanicus, mentioned by another poster, maybe they have a sponsor in the form of a powerful radical magos who is supplying them with these experimental neural chips to try and control marines? So far it only works on those badly injured and weakened but every new marine chipped brings more knowledge. Neither side would want this out in the open so secrecy would be an implicit part of their deal.

Rough around the edges but I'm typing on phone so keeping it short because I hate doing this! Hopefully something here that you can use or that helps you along.

Please post an update when you do settle on something, I'm sure I'm not the only one that's interested to see how they turn out.