I don't know much about modeling-agency headshots, but I've done a lot of them for dancers. I suspect that when they say they don't want a "portrait," all they mean is that they don't want it too fancy or "artistic." The equipment you've rented should be fine -- I don't think they're going to say that a moderate tele is too "portraity."
I've been told that the main purpose of a headshot is simply for identification -- they want to be able to match the name with a face. A second purpose is to let the customer size up at a quick glance whether the face is round or long, the nose is straight or bobbed, lips tight or full, etc. Often for a particular shot a specific facial type is needed, and the headshot should make it quick to determine whether or not the person fits those needs. There's no point in trying to make the headshot "sell" the model -- if they need somebody with a high forehead and your model has a low forehead, it's no dice no matter what you do. All you can try to do is make the person look professional and appealing, and give a good, accurate impression of what s/he actually looks like.
I'm attaching a few of my dancer headshots that have worked well for the people who requested them. They look pretty similar to model headshots I've seen, other than that the model headshots often have a pure white background (I don't think this is a requirement, but it's clean and simple; I put a little tone in mine to make them easier to reproduce in a program booklet, but that wouldn't be an issue for model headshots.)
As you can see, the shots are very simple:
-- I keep the background plain (shot #4 is an exception, but even there the pattern is subdued.)
-- I have the subject face almost straight into the camera, but not quite, so you can see the face shape, but don't lose all dimension of such things as nose and cheekbones. (Shot #1 is a slight exception here; this woman has an especially nice neck and shoulders, so I wanted to make sure to include them.)
-- I use a simple lighting plan that gives the face a bit of modeling but doesn't plunge anything into moody shadows. (Your ringlight may work fine, but I suspect it may be too flat for a face against a light background.) My usual lighting scheme consists of a big soft light slightly above and to one side of the camera; another soft light off to the side of the subject, to fill in shadows on the dark side of the face; and sometimes a hair light above and behind the subject. For women I'll often ask them to hold a white card flat below the chin, just below the camera view; this bounces some light up under the neck and nose to make the face look lighter.
-- I always try to show people smiling because we want them to look pleasant, and in your case the client may want to be able to have a good look at their teeth, so I'd suggest you shoot at least some smiley ones. If the subject doesn't want to smile because s/he prefers to look more serious or tougher (this is usually a problem with men) go ahead and make 'em happy, but shoot some smiley ones as well.
Again, I don't have much experience shooting for the modeling field, but I've done headshots that have helped people get jobs in dance and theater following this approach; it's pretty conservative, so you shouldn't go far wrong with it to start out, and if you find out that the modeling agency prefers something a bit different, you'll have a starting point that at least looks professional and won't hurt your model's reputation as a serious worker. Good luck!
