score:4
Turns out there's no direct easy way. Best way is to get a ticket to Beynou, and change an hour later for the train to Kungrad. From there local Uzbekistan transport will get one closer.
I ended up getting a shared taxi from Kungrad to Nukus (10,000 soms), then decided to go via Khiva. Nukus to Khiva was 25000 soms in a shared taxi, and from there bus and taxi options are available to Bukhara.