Unless you plan to race the bike, I would not use the Barnett pieces.
The springs are stronger. For a street bike, that means greater left hand fatigue.
I found the Barnett frictions were very hard to adjust to get both no slippage and no drag. Finding neutral when hot was always difficult.
I just found this thread. But, I was thinking as I read it that you were not adjusting the clutch correctly.
However, now that you have the cover off, you can see how the release mechanism works.
The arm on the outside turns a cam for the release on the inside. Make sure the marks on the outside coincide with the cam ramp engagement on the inside.
The outside adjustment merely positions the actuator arm close to the clutch rod, so the actuating cam has the greatest effect.
Once the cover adjustment is performed, then use the cable adjustment to get the lever in a comfortable position to operate, while still getting full disengagement.
One thought. To help with adjustment after assembly, mark the outer actuation lever to coincide with the internal cam position against the inner clutch actuating lever. You will be interested in the minimum and maximum cam points, as those will be the full engagement and maximum disengagement of the clutch disks. However, unless the actuation cam is worn, adjust the clutch just at the edge of full engagement, ought to get you full disengagement, using full clutch lever travel.
This is hard to explain, so my apologies if this writing is too difficult to understand.
I did compare your clutch disk pics to my Honda pieces (new parts with Honda numbers on the bag). And, they don't look the same. So, you may already have a Barnett clutch in your possession. But, I can't recall from the visual if that is what you do have. It's been years since I saw the last one.
I can take pics if you feel it necessary.
Lastly, the stack height of all the disks is important. There are two different kinds of steels and two kinds of frictions in the Honda pack. And, this height will interact with the clutch engagement adjustment. If the stack is too small, you may be outside the range of the Honda adjustment.
BTW, I have a 74 CB550 with original clutch and 42,000 miles on it. It doesn't quite grab like new, any more, but it still works. I certainly abused it in the late 70s and 80s.
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Cheers,