Putting a Face on Theatre
Your best bet is to avoid making this about whether the TD was right or wrong about his assertions vis a vis the lighting inventory. Bring your plot to a meeting with the building director, and discuss how much of what she wants is possible with the fixtures that are available. It's rarely possible to achieve everything that someone wants anyway, so it shouldn't be a surprise to the building manager that you can't accomplish everything she's asking for. Have some suggestions for a couple of different paths to take to get closer to achieving her goals for the space.
In terms of demonstrating your concerns, a couple of section drawings (a centerline and a transverse) that include visual representations of beam angles might work nicely to demonstrate the photometrics without making it seem like a bunch of mathematical gobbledygook.
Good luck. Just remember to keep the focus on the lighting, and not on who says what about which fixtures.
I'd put together the best plot I could with what is on-hand, and then put together some different options of plot's you'd recommend if there is an option to rent, or purchase new equipment.
Also putting together a short list of fixtures that you'd recommend they add to their inventory (based on expected use, size and type of show, etc) may be well accepted, as opposed to saying, "I think you have the wrong inventory...you should have this instead".
This way, you can educate, without alienating the people that made the initial purchase.
We don't have enough information to offer much useful advice, Your suggestion of a plot that will get the stage lit with the instruments they have sounds like the right approach. Developing a supplemental plot that can be filled using rentals or purchases as funds become available.
Any new purchase recommendations should consider the use of LED based fixtures. If there are a lot of dimmers and wiring in place, LEDs can be a hard argument to make due to the expense. However, if you have to consider buying a rack of dimmers and paying to run circuits out to the lighting positions then the LEDs start making a lot of sense. Most LEDs just need a typical wall outlet for power, and a DMX feed that will save some money.
Savings on electricity, heating cost, lamp replacement and gels will take a long time to pay for the extra expense in a theater, unless you are doing a lot of shows. LEDs do make sense in an environment where the lights get left on most of the day as in a store or office.
Another thing that bothers me about LEDs is that they often claim that the LEDs will last 50,000, but does that apply to the rest of the circuitry in the light engine. I may be wrong, but I got the impression that LED fixtures have "No user serviceable parts". Does that mean that a "burnt out" light has to be sent to the factory for repair in hopes that they still stock parts? There don't seem to be standard replacement "lamps" for these units at this point.