Mine was the same when we got her and still kind of is. One big thing, she would never “relieve” herself in the presence or under the watchful eyes of others (humans and especially other dogs!) Have you tried not watching?
Also habitual pee time ended up working for her... fed in morning, poop and wee time... fed at night, poop and wee time... before bed... wee time, we had to leave her out and use the word “wee” to encourage her... (she picks up words/ names/ actions really well)
Some days I come home to let her out in the middle of the day and she just stands there staring at the daisies growing...
My dad gives her a “pee” treat, but it can take rescue dogs a while to settle and feel comfortable....
Maybe use a command word if you see him wee so that when he needs to go you can communicate? Remmy will always tell us when she wants something and we go through her list, “do you want to wee? Poo? Drink? Food? Cuddle?” Etc and she will do the tippy tally paw dance to let me us know the correct option...
But mostly give it time. Basenji mixes are odd enough, but really wonderful! X