You don’t have to move your game to Unity, you could also just fix whatever the problem is in the plugin. You do get the source! (Which is not the case in Unity, btw.) That seems like less work than moving an entire game over.
Also, embedding web browsers really isn’t a good idea in the long run, because of the frequency by which bugs and holes are found. You’d need to keep updating your game every month with security patches to keep your users safe from potential threats on the greater web. A self-contained game doesn’t have this problem, because it doesn’t access new code that could contain new threats.
And, even if you do keep up with security updates for a while, at some point, you will stop, for whatever reason, and your users will be vulnerable and exposed.