I think the briefest and most true (if not the most helpful) answer regarding a penalty/benefit for Monroe via ED vs. RD is “there isn’t a clear signal either way”. As @lkbtnc noted, the merit scholarships page says “all applicants (both Early Decision and Regular Decision) are automatically considered for these merit-based awards”, and also that notifications don’t happen until March. (So even if you do apply and get in in the fall, you might not find out about Monroe until the spring.) I totally understand where you’re coming from on wanting to know about the odds, though.
I know a few people who got Monroe while applying ED. I know more people who got Monroe while applying RD. It’s worth noting, though, that more people apply RD, so don’t read too much into my “few” vs. “more”. More important, though: I know many more people who didn’t get Monroe at all, but who went to W&M and had an absolutely incredible, life-changing time there. As I mentioned in a post a few years back, “As cool as the Monroe program is, I really want to emphasize that the best opportunity before [you] isn’t Monroe, but W&M itself.”
I think this is the kind of situation where you won’t have perfect information and you’ll have to make the best decision for you with the intel you have. For better or worse, a lot of decisions you’ll run into in life from here out end up with that general shape — imperfect information and a binding commitment. I think you’ve done a good job of gathering the info you can, though, and think both paths in front of you have merit, as it were.
Seeing as you can’t know, I’d suggest maybe setting the Monroe to the side and weighing the other factors, like whether being done in December would be welcome. Especially if wrestling with mental health and motivation is a part of your story, I think an early acceptance would be huge. And since I haven’t seen you name any other schools that make your heart sing, I’m inclined to nudge you towards EDing to W&M. But ultimately, as you know, this will be one you’ll have to work through and make the best call for you.
All that is a long way of saying that I think it’s easy to spin out thinking about how to min/max the application process (or even just one small part of it, like a particular scholarship), and I want you to know that it will all work out well. You’ve done great work and are going to have an amazing college experience. I know it. I know it! Good luck with the decision. I promise it will all work out!