Without having explored the page and the sitemap, I would say that what your developer says is entirely possible and represents a good way forward from your situation.
Google have made a shift in the past few months with the way they index sitemaps - and if everything more or less matches up, even if they can't see the video on the page, they will often ascribe the page with a rich snippet and index the page as if there is a video present.
It's not surefire and having the video hidden beneath JS and delivered dynamically can prevent content being indexed - but it's certainly worth giving it a go (especially if you're using Wistia, as it's a simple process) and seeing if the video content gets indexed anyway. If it doesn't work, then I'd recommend going back and seeing if you can find a way to make the embed visible outside of the JS (perhaps just hide it with CSS and make it visible in a separate tab?). You could then have two methods of triggering the video playback for users if you wish.
Cheers,
Phil.
