I wanted to respond since I’ve had almost the exact same conversation regarding my “Imagining American Indians in Film” course. First of all, stick to your guns, as the committee is, in perhaps overly-simplistic terms, incorrect. It is a terminology in flux.
With that it mind, here are some concrete ways to respond.
I start by framing the conversation around the right of groups to choose their own terminology. This is written into the inclusive language policy at my institution, so I make sure to point to that.
In that sense, I always use national/tribal names when possible, deferring to the term preferred by the nation itself. For example, Dine rather than Navajo or Haudenosaunee rather than Iroquois. In moments when I need to refer to people more broadly, I use American Indian. On this I turn to a number of sources to defend that position. The first are American Indians themselves. While the opinion is not unanimous, high-profile individuals have made the case for keeping American Indian. Sherman Alexie is a particularly popular one. There are also larger connections to the history of American Indian activism that many Indian people want to preserve. Finally, there is a legal case for continuing to identify as American Indian, as this is the term enshrined in case law.
Sorry if this is more than you were looking for, but I thought it’d be helpful to share my own approaches.