Not knowing your exact route, but as described, it seems to potentially traverse the Great State of Texas IN WHICH OPEN CARRY IS PROHIBITED. That is not to say you will have problems but if you did, your CO may be just a tad bit perturbed!
Also, per my understanding, without a helmet if you were to be injured YOU could face military prosecution for the damage you imposed upon military property (YOU).
Enjoy your ride, carry as able and where legal. Have fun and don't get into trouble!
No requirement that I am aware of to "ziptie" your action. For the federal safe law, the handgun just needs to be secured in a case and preferably NOT immediately available to the driver. The Communist State of California is the only one I am aware of at least until you get to Texas that will restrict your ability to carry your firearm fully loaded (this applies to the firearm, NOT THE OPERATOR)! Texas says no open carry but does not care if it is loaded!