Is reproduction of copyrighted material in a research proposal fair use? Say that I am writing a grant application to be submitted to a funding agency. The application is not a public document and so it will be only accessible to the review panel, the collaborators involved, and administrative staff both at my institution and at the funding agency.

Now imagine I want to make a point, in particular regarding the wider interest of my proposed research topic, for which a figure that appears in Nice Paper by Reputed Author is a perfect fit. Unless Nice Paper has been published under some Creative Commons-type license (assume it hasn't) copyright in principle prevents me from freely using this figure in my proposal.

However, one could argue that if 1) I credit Reputed Author for the figure and reference Nice Paper in my proposal, and 2) given that I am not using the figure for commercial purposes and that only a very limited amount of people can access this document, then 3) it is reasonable (and it constitutes fair use) for me to reproduce the figure without seeking explicit permission. Getting explicit permission could potentially be a tiresome process since it's probably the journal that published Nice Paper that holds the copyright.

Am I right to assume that reproduction of copyrighted material on a research proposal with the characteristics described above constitutes fair use?
A proposal is not a publication, and as such you do not need to worry about the usual strictures of copyright.  It doesn't really matter what the technicalities are: if the proposal is to remain private, de facto it is as though it doesn't exist from the copyright holder's perspective.

Academic honesty, however, is another story.  If the figure is not yours, you absolutely must give credit to its actual author.  Further, you need to ask yourself: is including somebody else's work actually good evidence for my ability to execute the work that I am proposing?  In some cases, it may still be the right thing to do, but usually if it is a diagram, you will probably do better to create your own independent version, and if it is data you will probably do better to cite the result but not to waste precious proposal space reproducing their figure.