This is all very odd, but I don't really think DH has handled it well.
If his job is recruitment and his bosses want him to recruit a certain person then I think it is important that he actually contact her (not just send emails with details) and get some sort of response from her. I would expect a recruiter to call or send a detailed email requesting a response and to follow up until they get a yes or a no.
I understand what you're saying. In this case, the bosses told the staff to get Tina on contract. No one knew she hadn't been approached before. Usually, when the bosses tell them to get someone on contract it's because the bosses have taken steps to recruit the person already, and the person has been promised the opportunity. That is way outside standard procedure, which frustrates DH and his staff, so what happened in Tina's case is even farther outside of standard procedure.
The bosses have the authority to violate procedure, but DH and his staff don't, so if the bosses want the staff to do something outside of procedure, I think they need to be more directive about it. In other words, if DH did something like this independently, it would be a procedural violation, and unacceptable. The bosses likely will continue to circumvent procedure, so DH will have to adapt.
He tries so hard to tow the line, and it's hard when the bosses move the line like this. He has to enforce proper procedure, so this kind of situation makes it hard to do that. He has had to learn that doing the right thing is relative to what the bosses want. Before, DH's role in enforcement of procedure was uniformly respected. There was a significant leadership change (I mentioned culture change in an earlier post), and now the environment is much different. Enforcing procedure is now random and unpredictable. DH just tries to do whatever the bosses want.
If anything, I think a lesson DH has learned from the Tina situation is to clarify with the bosses whether the people they are referring are already aware that they are being recruited. If not, his assistant can adjust her approach. It's one more random procedure change in an increasing list of random changes.