This is really interesting - an example of how something that in many respects is an advance in technology (digital over analogue) can come back to bite the unwary.
It's a recognised problem in concert halls around the world, so much so that many professional settings and TV broadcasters (Broadway theaters, Sydney Opera House etc al) have analogue video cameras of the conductor feeding to analogue CRT monitors. And a shed full of old CRT tellys round the back as a space for when the monitors break down (I don't even know if anyone still makes new CRTs). Samsung and probably other companies do still make analogue CCTVs for this reason.
You can get professional broadcast standard monitors for a price (a few thousands of pounds) which boast typically less than 1 frame per second latency - but at 25 frames per second that still amounts to admitting up to 40ms latency!
This is a real problem for avid computer gamers too as they need the picture on the screen to update as soon as they move their joystick. Gamer forums are full of discussions about which monitors have the lowest latency. Interestingly it can sometimes be the cheaper ones (since they have less advanced video processing to articificially "improve" the quality of the image being received from the camera). Some monitors have a "gamer" mode that disables the processing. (Conversely some upmarket audiovisual amplifiers have a built in adjustable sound delay so that you can slow your sound down to synchronise with the delay caused by your expensive TV's built in video processing that slows the picture down - an expensive solution to an expensive problem!) Some gamers advocate connecting a source directly via HDMI as digital out (in your case from a camera with HDMI out) to digital in should mean no additional processing of analogue to digital images (especially if the onboard processer is disabled). Others say use VGA or component out for certain monitors since if they can detect an HDMI signal and assume you want it to have additional processing!
If you use off the shelf monitors there are additional considerations. There are two common types of LCD monitor, TN and IPS. TN is cheaper and has a faster response time (which is quite different from latency - it's how long an image persists for on the screen after the source has changed). So TN screens are better for gaming - fast movement can appear blurred. However, they have quite poor viewing angles other than viewed from head-on. If you are sitting at an organ bench and moving your head around, the colours of the screen will shift depending on which angle you are looking at it from. That's not the case with an IPS screen - but it might be slightly more blurry when the conductor is waving their arms around. Ideally you would want an IPS screen with minimal latency and minimal response time - but they come at a hefty price. OLED screens are a third type but are also in the very high price bracket.
Frankly your cheapest "best bet" is perhaps to look for secondhand ex-broadcast equipment on Ebay. Failing that if you have the time to try out or seek out off the shelf combinations of camera, cable and monitor that work for you great, or pay for an expensive pro-grade set up. Make sure if you go for professional shopping advice that you explain why you need as close to zero lag/latency as possible. And if the shop assistant doesn't know what lag is, or confuses it with response time, go to another shop.