I am in a similar boat, work for a state university. I do not get travel orders. We organize our own travel, I have a travel budget. I assume my university ID would be sufficient?
Regarding the fine print "select the State Government rate when available": May I select the only available Government/military rate, or may I only book if a state government rate is available?
Should be fine either way.
I still use the rate for leisure, and never get bothered (rare ask for id). If you're actually traveling on business, you should be less concerned. If in a very rare instance they deny you, despite being on government business, you could probably take it up with corporate and get it sorted out.