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Here are my examples.
One screenshot will show how my Base Commander perk is set up. Another will show me at bat with the perk active. Another shows when my teammate at bat, and the Base Commaner perk is no longer active. The final shows the Rattled perk active with that same teammate at bat.
I see what you're saying, and I apologize if I'm not being clear with what is happening.
If the BR perks are supposed to be persistent throughout my time either at bat or on the bases, then it definitely isn't persisting when I end up on base (again, just regarding the BR perks specifically) and hit pause to view what's currently active. It wouldn't make sense for the BR perks to activate while I'm at bat since I have no use for base stealing if I'm at bat.
I get the Rattled perk to activate when I'm on the bases, though, even though that's a "Batting" perk. But I'll have the Base Commander perk "activate" while I'm at bat, but not persist or activate when I'm on base, when the primary perk for that is a stealing speed boost (it also does not show as activated in the pause menu).
I'll try to grab a screenshot or clip soon. I may be misunderstanding the process for the perks, and I'm happy to be corrected. It might very be bad UI design rather than an actual bug, or I could be crazy, or both. 
@broken_toy1_PSN They do not show up for me when they should be activated. As another poster mentioned, they'll show up while I am up to bat, but stealing boosts don't really help in that moment of course.
Will there ever be any acknowledgement of baserunning perks being bugged? Or any updates on potential fixes in progress for these perks being bugged?
I'm sure folks can only report bugs so many times before just giving up without seeing any mention that devs are aware of certain things. I truly want to try to stay positive/hopeful, but it has been months and it's becoming incredibly discouraging.
I have also noticed plenty of bugged challenges. The most recently example: I just had the "get an out with a pitch out of the strike zone" challenge fail with a batter grounding out a pitch outside.