what do i have to do to convince regis to switch sides in the "oh my papa"quest?

Assuming you have already talked to Papa Khan in the Longhouse, and Regis has approached you upon leaving:

He asks you to get the approval of PK's other Lieutenants- you'll need to talk to them and convince them it's a good idea.
If you have not yet done Aba Daba Honeymoon, Diane may ask you to do so before agreeing; I always do it beforehand, so I could be mistaken here.
You'll need proof of Karl's shenanigans in order to get Regis to go along, this can easily be stolen from Karl's locker when he's not in his room (stay [HIDDEN] while doing so, just in case).
Once you have convinced all four (including Regis), you can then talk to PK again. He will always refuse this first attempt, but in the process Karl will be killed if you have the above proof; this is important, as Karl has to go for this to work.
After Karl is killed, talk to PK a third time. You will now have two [Speech 75] options; either one will put him in favor of the idea, but you'll need a second check at 75 to finally convince him.
Alternatively, if your Speech is not high enough you can appeal to PK's reverence for the tribe's traditions; you'll be given two options for what to provide him with that will convince him.

IMPORTANT: in either case, one of the two options will result in the Khans allying with NCR while the other will convince PK it's time to go elsewhere; which is better depends on what you're trying to do with that character.