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?
Even many U.S. government employees do not get "travel orders". I really think it's a misunderstanding from Hyatt than how civilian government travel works. The only time they ask for "travel orders" is when I am on official travel to a state that does not require U.S. government employees on official travel to due tax on hotel charges. A printout from Concur with my agency's logo has always worked.