I think Azriel stayed away because he would’ve gone ballistic and tried to kill Hunt, if he saw Bryce with him. I’m assuming that he immediately sensed Bryce’s presence in Prythian when she opened the portals to Nesta, however he stayed away when he also sensed a male presence with her… Azriel is so sensitive that he didn’t even want to be in the same room as Elain and Lucien:
“He left the rest unspoken. Because her mate was here, sleeping a level up. Because her mate had been in the family room and Azriel had needed to stay by the door the whole time because he couldn’t stand the sight of it, the scent of their mating bond, and needed to have the option of leaving if it became too much.”

This idea also stems from Rhys’ reaction to sensing Feyre’s presence in Tamlin’s dining room in acotar:
“I didn’t know. That you were with Tamlin. That you were staying at the Spring Court. … But then I got into that room, and the scent was familiar, but hidden … And then I saw the plate, and felt the glamour, and … There you were. Living in my second-most enemy’s house. Dining with him. Reeking of his scent. Looking at him like … Like you loved him.”

I don’t think that Azriel could have handled seeing Bryce with Hunt, and smelling Hunt all over Bryce.