I'm going to be "that guy" and ask just a few things first ...
1) how are you defining success here? What does "need cleaner oil" mean?
2) have you attained UOAs? Do you have any idea how much or little soot and fuel you have, or where your wear rates are at?
3) what criteria have you set out as a means to compare/contrast the different oils you've tried, and those you still want to try.
I mean, you've apparently been unhappy with several, or you'd not still be on a quest. So please tell us what quantifiable, measurable characteristics you've been dissatisfied with so far. If you've not done these things, then how in the world are you affecting judgement on any lube choice?
As Charlie said, no oil is going to stop or alter regens; only your operational pattern can do that. If you're getting so many regens, it means you're not getting the engine hot enough in normal operation. Why would the OCI duration have effect on the regens, anyway, in your mind?
CJ-4 lubes are made with high soot in mind. No lube (yet to my knowledge) has been developed, as a lube classification, to specifically deal with high fuel content.
Until you define your perceived issues much more thoroughly, and then use UOAs to compare/contrast lube performance, you're just "feeling" your way through this. And that's OK; there's nothing wrong with that. But there is not one shred of "help" we can offer because it's hard to see emotion on a UOA ... You're not alone; LOTS of people do this. They "feel" (aka "want", in this case) something to happen, but they have no idea how they are going to successfully measure the competitors or quantify results. Hence, it's all about opinions and not about facts.
Your choice; either feel or think your way through this.