I think this will depend on the specific property. In theory, the government rate is for "authorized travel" only, so it would only apply if your wife were traveling on orders, in which case she'd be the one booking the room and paying. In a situation like that, you could ask the hotel if they'd honor your Gold status since you're also staying in the room (and many properties likely would).

In practice, many hilton properties allow the rate for leisure travel, and I suspect that being able to show your wife's military ID would be sufficient to justify use of the rate if anyone asked (which they rarely do) no matter who had actually booked the room under their account. If I were in your situation I would email or call the hotel in advance and check with them to be safe, so that you don't show up at the property and run into any issues.