I've looked into this, and most of these inexpensive systems are composite PAL, so you'd need to convert HDMI or SDI to composite PAL in order for it to work. If you want to use your normal production monitor on the receiving end (instead of a purpose-built FPV monitor with a built-in receiver), you might also need another converter if your monitor can't accept the composite signal natively. Many of these systems operate on 2.4 or 5.8 GHz, so you may experience interference if a Teradek (or similar) is in use in the area, or even just a lot of WiFi. I almost went this route, but it just seemed like I would need so many pieces (and points of failure) to get it working, and that the end result would likely be somewhat clunky.
Amimon (the company that provides the chipsets/underlying tech to Teradek, Paralinx, Vaxis, Arri, etc, and was recently acquired by Teradek) makes an HD wireless video system called the Connex that's targeted at FPV use, but does a fantastic job for general purpose video as well. Quality/range is on par with Teradek (it's the same chips inside), and I can easily get 1000 feet with standard antennas on my set. With a panel antenna, you can go 3-5k feet. Unlike a Teradek, the Connex won't transmit the embedded audio, but that doesn't sound like a problem for this application. The whole system is HDMI-only, but most monitors will internally cross-convert, and there are lots of compact converters in case you can't find an HDMI loop-out. Power-wise, the transmitter uses uncommon connectors, but if you get a DTAP cable, a LEMO cable, and a wall-wart, you should be set for most circumstances. The receiver just has a standard barrel jack. Both transmitter and receiver have very wide voltage ranges, since it's meant for RC use. The transmitters are tiny, so it should be unobtrusive wherever you stick it. I'm not sure how the receiver does with RF spray (mostly use my system for aerial/gimbal work), but now that I'm curious I'll definitely test it out. Sets can be had for ~1k used on eBay, so a pretty good value for the performance when you compare to "film industry" brands. There have been several versions, so the best combination currently is a Connex Mini Air Unit (the one with SMA connectors), and Connex (not mini) Ground Unit (the one with 5 antennas), more recently marketed as the Connex Fusion Ground Unit.