Watching Billie on NG, she seems very nervous and scared to me, and I would be, too.
I do think there is a method to their madness, though. By telling Billie her BF failed, they figured she would either get suspicious of him and ask him to leave... or she would go into protective mode and stand by him. When she threw him out, that tells LE that she will cooperate with them, and that she cares more about finding her daughter. If she had clung to the BF and made excuses for him, then they would have started thinking she had something to do with it or at the least, knew he did and was protecting him.
This is what they do... get the suspect out of their comfort zone, make them feel like the ones they love are turning their backs on him. The next step is to see what he does now, and I'm sure they are keeping a close eye on him.
All JMO, of course.