My boss takes "criticize in private, praise in public" to the extreme, and it's exhausting. What do I do?

I work as an engineer on a small team in a small company. The difference between what I hear from my manager in our 1:1s vs. what I hear in a larger group is huge.

In front of company leadership, I hear "Redditor is great at taking lead on projects." "Redditor would be a great manager, can she manage the incoming intern?" "Redditor worked really hard on X project and did a great job."

In our 1:1s, I am desperately fighting for chances for visibility. For most of this year, I've been working directly with other teams and strategically planning my work without my manager. In such a small company, this has made a huge impact and been wildly successful (from previous performance reviews). My manager started being more involved this summer. Now, I'm battling to even have the privilege of making a Gantt chart in a project page for a plan and project that I came up with and am the only one working on. In project meetings, I prepare written words for him to present.

When I pushed my manager in our 1:1s for a reason recently, I was told that my technical skills aren't up to par. Look, if a PhD in a technical position for two years still hasn't mastered technical skills, then this ain't the job for me -- so I scheduled a meeting between me, my manager, and his manager for a second opinion. In that meeting, my manager continued to sing my praises. Privately, I'm still being denied any chance for higher-level work or visibility.

I'm now at the point where I'm starting to record our 1:1s. I plan to record what exact technical skills I'm not being successful at, preparing a document to justify how I've been wildly successful at them, and sharing with him and his manager to align.

This is all just... so exhausting. I don't have time for this, his boss doesn't have time for this, and I just want to continue doing great work.

Of course, I'm probably being managed out, but I have no damn desire to leave the company. This would result in a significant pay cut, and also a huge career setback for me.