According to this website, they recommend 50-65% rim width to tire width. For 35mm tire, that would be 17-23mm.
13.7mm rim is probably for 20-28mm tires.

The rim you showed is probably Kinlin XR240.
You could probably look at Kinlin TL21, which is 21mm, or Jetset CH240 which is 18mm (surprisingly heavier than the Kinlin).
https://www.ginkgo-veloteile.de/gink...8-Inch-355-mm/
They write on the website that shipping to the US is 37 euro, probably DHL. However, if you used air mail, I think it's around 14 euro to be delivered by USPS.
These prices include 19% tax, so the Kinlin TL21 would be 38 euro without tax if exporting out of EU.

Edit: oh, the Kinlin TL21 that they have seems to be for disc brake only. The english page has errors, it's not carbon.