Conference room microphone recommendation

So, here's the situation; The board of directors for our small non-profit has outgrown the boardroom they have normally been meeting in.

The decision was made to move the meetings into our larger exhibit hall, but that created lots of issues with background noise from A.C. vents and difficulty hearing some of the board members who were furthest away from the webcam.

There's too many board members for a wired or even wireless individual mic setup for each board member.

I was looking at a ceiling mount directional mic solution like the Shure MXA920W-STA. But, that's a little out of our price range for one. And for another, I don't want to go through the trouble of convincing our facilities director this is something we need and he would need to install it.

Is there anything out there for a good price that could be placed at the front of the hall and still do a good job of picking up meeting members?