All of that info really helps. Thanks for that.
Both the GH5 and the BMPCC 4K are great cameras, and either option will work well for your situation. They each have their own pros and cons:
- Smaller body
- Flip-out and rotating screen
- Better battery life (for the gimbal camera)
- Built-in image stabilization (some of the best available)
- 4K HDMI output (for future or special projects)
- Not as good in lower light compared to the BMPCC
- Screen is smaller, so it can be more difficult for camera ops to see for focus
- Autofocus is slow and unreliable
- Requires a dummy battery adapter for AC power
BMPCC 4K Pros
- Larger built-in screen, big enough for most situations
- Has separate DC power port and power adapter
- Records in better quality codecs
- Can record to SD card, CFast card, or USB-C drive
- Can apply a LUT to only the HDMI output, giving you tons of control over the style and look
BMPCC 4K Cons
- Awkward shape body - difficult to mount on a lot of smaller gimbals
- Poor battery life
- HDMI output is only 1080p
- Autofocus is slow and unreliable, AND it is single touch only (no constant AF)
- No image stabilization - stabilization has to come from how the camera is rigged or from lenses that have IS built-in
As far as autofocus is concerned, you shouldn't need it for static camera shots at all. Once you set focus it shouldn't need to change. Autofocus causes more distractions and issues than it solves in these situations, that's why I recommend turning it off.
The BMPCC will still be able to do a LUT on the HDMI output. The ATEM connection is not necessary for that. Just so you know. The ATEM will only give you control of the focus, iris, record start/stop, tally lights when a camera is live, and maybe zoom if you use one of the lenses that has power zoom (only 3 lenses that offer that, and none of them I recommend).
As far as lenses, I recommend sticking with native Micro Four Thirds lenses as much as possible. You could use an EF to MFT adapter for your Canon 18-135mm if you'd like, but that's about the only reason I would recommend one in this situation. I would encourage you not to get caught up in the crop debates or getting the extra stop of light. A lot of people get into the numbers way too much where they confuse themselves (and others) or they overthink it. For your situation you really don't need a speed booster.
You can see my lens recommendations and some ideas about what types of framing you can get from certain distances here: http://joelwsmith.com/live-streaming-cameras#micro-four-thirds-lenses