Camera w4k Internal + Continuous Autofocus + Clean HDMI Out?

Hello, I'm looking for cameras recommendation to use with my Atem Extreme ISO that can do the following:

- Internal 4k 24/30p unlimited recording
- Clean HDMI out while recording (1080p is fine)
- Continuous Autofocus
- Around $2000-2500 USD

If the BMPCC cameras could do Continuous Autofocus this wouldn't be an issue -- but I often film speakers and musicians who tend to drift forward and backward on stage, and also low-lighting scenarios where I'd like to be around f2.8 -- and I don't want to have a camera operator on all 6-8 cameras pulling focus.

I'm worried about a Canon R6 / Sony A7 IV overheating while recording 4k Internal Recording while also sending out the HDMI feed to my mixer. I saw some Prosumer camcorders, but it's hard to tell on spec sheets if the cameras drop any features while internal recording -- such as clean hdmi, autofocus, or the screen on the camera.

Several of the Panasonic mirrorless cameras are capable of this, including the GH5, GH5S, GH5 II, and BGH1 (Micro 4/3) as well as the Panasonic S1 (full frame). Panasonic's autofocus isn't quite up to the level of recent Canon or Sony cameras though, so you may have to be a bit more careful about what situations you use it in.

You might also look at some of the Canon camcorders with a 1-inch sensor and dual pixel autofocus, such as the XA50, XA55, or perhaps even some of the more consumer oriented models, such as the GX10 or HF G60.