I’ve never really delved into those myself and not sure how much help I can be. However, I would think a question like this might be best directed to the host provider because it’s actually their servers that will need to sustain that capability. Also depends on how many will be involved in your webinars. You’ve already started looking into the issues relating to parent/child themes…which is a good starting point because I had no idea they would create issues.
For plugins and scripts that do webinars, the only thing I can suggest is search on Google and read reviews, ratings, comments (especially comments), check out what support is available, and definitely send the developers questions before you decide on one.