I can't see your link, either. But I'm going to jump in and give you some photos anyway, in the hopes that they will help you out.
Math is going to be your friend as you figure out what to do. You'll have to be careful if you want your band to sit low and tuck under the prongs of your e-ring for a flush look. Even just a half-millimeter more or less on either ring can make a difference.
The top set of photos are my rings that sit flush (or so close that the gap isn't noticeable in real life). Both of these e-rings are pretty wide and pretty tall in profile, creating a space for the band. The 3mm wide band is Unique #7796, set by my local jeweler. The 4.5mm wide band is Unique #7102, set by board vendor Amcor. (FYI: MoCo sells these settings as SOL224 and 222.) The bottom set of photos are a routine petite tiffany solitaire that I bought off eBay used.
The anniversary bands in the first two yellow gold sets are both low profiles. I had to shop carefully to keep them at 2.5mm high. The eternity band in the white gold set is more of a standard height at 3mm high, and I personally think it overwhelms this dainty e-ring band. And no way are those two sitting flush, period.
Sharp mind, soft belly.