Hi there,
I see that you haven’t received an answer yet so I’ll have a go in case you’re still interested!
The problem about including Championship data is a bit tricky indeed. The issue is that Championship teams will generally not play against Premier League teams, except if they’re promoted or relegated, right? That’s not great since there won’t be a lot of mixing between the levels. Still, I think you could just give it a go and see if you get a reasonable result.
Beyond that I’m also not super sure. One thing you could try is to modify the prior. I’m guessing each team’s strengths is drawn from some distribution, probably with the same mean and variance? If so, you could draw the Premier League initial skills from a different distribution, maybe with a mean that is strictly larger to indicate that they’re better.
Anyway, hope this is helpful – curious to hear what you think if you’re still working on this!
Martin