Indoor Video Capture

Terfyn

All cams can be fitted with a wide angle add on lens. I had  a Neewer 0.45mm wide angle for my HC-V750, this gave an equivalent of 0.15x without viginetting. The V800 has a 62mm lens mount so not sourced a wide angle for that one yet.

12harry

You room dimensions are putting filming under some pressure - is this because the room is limited, or that that's where you have been given, perhaps?
Conversion lenses do introduce some measure of distortion and may cut corners too. However, if you find one that works OK for your needs, that's the solution.
You can get close to the WA need (or not) by filming the stage with/without the conversion lens....OR you can take a still using a compact camera with a known WA spec, then relate that to the camcorder you wish to use.  i.e. Try before you Buy.
Could it be your friend has extensive experience of these PTZ cameras? ( They're never mentioned here, AGAIK.)
A ( pair of ) modest camcorders could be a better use of Funds - but you've not said "what" you are filming - Especially if one is not so wide, as this is rather easier for the audience to follow - Wide-angles mean their attention is not directed. You can zoom-in at the Edit stage, but this shows up definition loss and focus errors. Even if you have only one "position" it is a great help to the Editor to have different shots available. This may put pressure on the Operator, but the WA can be left alone as defects won't show much - However the "better" footage will need the zoomed-in camera to be more accurate - so the Editor has the WA to fall back on.
As I may have mentioned the use of lighting should be to direct the audience's attention to the Speaker/Product(etc.) and it also helps to allow the background to be less distracting.  Out eyes are poor judges of lighting ( due to our brains' ability ), but the background should be visually obviously darker than the subject(s) - and this may mean having some control of light-positioning - rather than lighting as I understand you suggested.

OptiMystic

I am limited by the size of the studio.  Here is some footage of the space using an iPhone with a clip lens.  Using airpods for the audio didn’t help either.  Https://www.bit.ly/2EAPVX6

Terfyn

So I presume you are filming golf swings.
Returning to my suggestion of a camcorder, the 800 will easily take in your subject PLUS it has a slow motion feature which, I would suggest, is essential for golf training. In fact this camera range has been used by others for exactly that purpose.
The use of a wide angle add on lens does not cause the problems suggested above because it is rare that you would use it at the full wide zoom on the camera. These lenses do show signs of ""barreling" at the widest angle but moving the zoom in a bit solves this as well as viginetting.
Lighting is certainly not a problem from the shots shown.
You will get echo when using the on-board mics but it would be so easy to set up a mic wired to the camera for close up sound capture.

The suggestion of two cameras, one facing the golfer and one to the RHS is a good idea especially as most video editors can now sync the two shots together.

OptiMystic

I have 2 Point Grey grasshopper USB3 cams I use for video swing analysis so i can get 120fps; similar to the casio exF1 I use outside.  this camera does not have a throw capable enough to sit at the back of the room and get me swinging in the shot.   
I steered away from camcorders years ago (panasonic 3ccd was my last) because again, I could't get the frame large enough without a fish eye which made the whole view rounded

Terfyn

I think you will find it is very different now. At one time fish eye were all the rage. The 0.45x I use on my 750 shows no sign of barreling but slight viginetting at the widest angle setting. I often use the lens for inside shots and it is like increasing the room size by half again.

12harry

Many Sony camcorders offer a  "Golf-Shot" feature...this puts many exposures on one frame ( as I understand)....not sure if it's a moving image but it's probably not that important, since you can study each swing sliced into many pieces.
It resembles those multiple-flash photos way back in strobe-days, with film.

Not sure how the lens-angle helps  - and you'll need to trigger the shot using the ball-sound . . . Sorry, you can tell I've not used the feature on my Sony.... but I often use the 5x filming Slo-Mo .... but I understand Sony has dropped this for their latest camcorders . . . . 4K tech. may be the reason.
Maybe worth getting a Demo . . . ?​Someone will have a sony camcorder.... and has probably never used the high-speed features.

Cheers

OptiMystic

Thank you for all your input.  For the actual swing capture, my dual UsB3.0 are great. I run them all thru a software called GASP and then to make a info/Vlog type video, I use a broadcasting software called Xsplit with 2 Logitech 922 and a yeti mic but I’m looking to upgrade to a better camera and a lapel mic, hence my post.  I’m checking out camcorders tomorrow for the right focal length.

OptiMystic

I’ve also tried a GoPro hero 5 on 2.7K @ 60fps using the Linear view and it’s a great image. Also connected the audio usbC cable for the wireless mic and it’s a step up from logitech (which has a slight lag)

OptiMystic

Anyone with experience with this:

Blackmagic Micro Studio Camera 4K | Blackmagic Design