The big thing is to make it clear that you’re not affiliated with BareMetrics – say by similarity in page design, logo, etc. Not only is that the part most likely to piss off Josh, it’s also the most likely to be actually illegal.
For instance, you’re not allowed to create a cola with “cola” in the name, the red can and the Coca-Cola font. That’s considered their look, and they have legally protected it, long since.
By using a different page design, color scheme and/or logo, you’re making it clear (and legally you must make it clear) that you’re not “Baremetrics for Braintree” in the sense of being the same company’s Braintree product.
That’s the legal requirement. The rest is, y’know, not pissing off random geeks, which you can put as high or low on your task list as you like. You only care if your customers feel the same way.
I don’t think you’re currently quite confusing enough that you could be sued for it, but it doesn’t help that you and Baremetrics both have very “generic current web” looks. A different color scheme could help a lot with minimal work.