The BrilliantEarth set looks like a good one.

Still a bit flashy, but not incredibly so.
One of my friends raised this point a couple of years ago and I took a look to see what the position is in the UK, the major retailers I looked at from the the largest jewellery chain in the UK to the equivalent of Tiffany +1 had an ethical policy on diamonds on their website.
Therefore I'd assume whoever you buy from should be able to confirm that the jewel isn't a conflict diamond.
Actually, those diamonds don't come with a guarantee with any sort of meaning, which is rather unfortunate.

Diamonds are really hard to track, and even when they're tracked to a mine that claims to be conflict-free, usually that's just what the mine claims. The review process is very flawed, and countries and mines are rarely, if ever, checked. I think it says so in your link to the Kimberly Certification (didn't have time to read it thoroughly, although I did read it ages ago).
Brilliant Earth (and several other places like it) are different because they're mostly mined in Canada, and I don't believe that wide-scale slavery is happening in open view in the Northwest Territories.
In most cases, I'd have no problem buying things made in Africa. If the money helps their economy, then wonderful. However, diamonds are different. Buying African diamonds doesn't actually help their local economies or the livelihood of the people. They get none of the benefit. You just get more slavery and war, it seems.
To be fair, I'm not incredibly well-studied in this area. Take it for what it's worth.....it's my opinion after doing a bit of reading a year ago.