HDMI encompasses a few standards. SDI is far more specific. Your HDMI computer monitors will be expecting RGB colourspace, in a variety of resolutions/frame rates. Your SDI adapters will send only video standards down SDI, then only video HDMI standards out of the HDMI - so 720p/1080i/1080p in YUV colourspace. TVs will display these standards, many computer monitors will not.
Unless you've got the cash for something like a Marshall SDI camera, I'd stick with composite. Most big musicals are still on composite for MD cameras (though there is now a growing move to SDI for FOH picture). Get a good quality composite camera (I recommend the Bosch cameras which can be found quite affordably on eBay), a composite video splitter (or distribution amplifier to use the proper term - again, plenty on eBay), use your existing BNC cabling, and either use cheap TVs (could be higher latency) or SD native screens - varying sizes available in varying ruggedness at varying price points.