DOD is already SpaceX's biggest customer. It's probably already in the weapons game, but that's not something you're going to hear about, because it would be classified.
Though their launch payloads are often classified, that they're launching DOD payloads isn't classified and given the contracts that are public as reasonable proxies for the ones that aren't, there's simply no chance the DOD's manifest is SpaceX's biggest in terms of launches or dollars.
NASA, for example, launches on SpaceX way more often than DOD. You can easily count the launches yourself by reviewing SpaceX's historical manifest on Wikipedia. I think there have been 6 or 7 DOD launches in the last three years while there have been 15 or 16 NASA launches in the same time period. Some DOD launches are a bit more lucrative than some NASA launches, but the inverse is also true.
You might go on to assert that SpaceX has other DOD projects like its Starlink/missile tracking and E2E cargo (and potentially troop) delivery (and even money for Raptor dev and other smaller programs like the long ago DARPA investment in Falcon 1) but it's clear from various reporting that those are tiny investments compared to a year or three's worth of orbital launches.
Yes, the USG is SpaceX's biggest customer. No, the DOD is not SpaceX's biggest USG customer. That title, for now, belongs squarely to NASA.
You can see, clearly, far more launches for DOD umbrella agencies than for NASA.