Technically, I don't think you're eligible for the Government rate. The rate rules at most properties have the following statement:
** Government Contractors are not Eligible for the Govt Per Diem Rate **
Note Government Contractors working on Government "Cost
Reimbursable Contract" are not Eligible for the Govt Rate
It would seem that they are specifically excluding folks who don't work for the Government, regardless of the fact that the taxpayer is the one who is ultimately picking up the tab. If you want to be bulletproof, have someone at the GSA or the agency that funds you guys write a letter on agency letterhead stating that you're eligible for the Gov't rate.
All that aside, I've found that properties often don't ask to see my Gov't ID when I'm checking in. Disclaimer: I usually stay at the same properties repeatedly, so that may have something to do with it.