Help w/ Countertop Dishwasher Connection/Adapter

I have a countertop dishwasher that needs to be hooked into my faucet, but the major obstacle is that my faucet is a pull-out spray head. I can pop the bottom of the spray head off to reveal the aerator underneath and I’ve purchased an adapter to try to connect the faucet aerator to the to dishwasher water intake, but the adapter doesn’t fit into faucet aerator and I’ve been struggling to figure out the sizing to find a smaller version. Any guidance on parts, sizing, and assembly for a viable connection would be much appreciated.

But more generally it would be great to have more advice about the best way to solve this problem generally. I read this blog post with one suggestion, but it just seems to janky too be practical. Any other suggestions?

Countertop Dishwasher:

Pull-Out Spray Head Faucet:

The pull down head with the black top taken off:

The other side of the black top that screws into the head plus a washer that sits in between:

Water Connection into Dishwasher:

Aerator that came with Dishwasher:

Adapter in Dishwasher Hookup (15/16"-27 outside threads + 55/64"-27 inside threads):