With the Mobilize order, the card seems to read that you can start a battle with the first activation, but not if you get a second from the Supremacy Bonus. A more detailed explanation in the rulebook is a bit vague, but seems to indicate you can only start one battle with the card, but seems to imply that it could be either activation that does it. So if I activate an area with neutral units and my diplomacy succeeds, and I have the Supremacy Bonus available, can I then use my second activation token to attack an enemy (or try to recruit neutrals in another hex, without being forced to retreat on a failure)?

Yes. If you have the supremacy bonus for Mobilize, you activate one area, and either start a battle or don't. If you don't start a battle with the first activation, then you may start a battle with the second. Essentially, the primary ability and supremacy ability are exactly the same, but you can only start one battle between them.