If you are not going to be using it at all for commercial purposes, then you should be able to register it as a Historical Vehicle. This will exempt you from paying weight fees (which are the bulk of that huge annual cost they quoted you) or needing to stop at truck scales. On the other hand, it places restrictions on what the truck can be used for; i.e., it must be used primarily for display and parade purposes. Note that according to the CHP's commercial enforcement division, the key word is "primarily", ie. 51% or more. The DMV clerk will probably tell you that it can only be used for parades and display. Smile, nod, and don't try to correct them. You will also be able to insure it through a collector car insurance agency like J.C. Taylor, which will cost a whole bunch less than if you are going to be using the truck for commercial purposes.
Carefully consider how you will be using the truck. If it'll be a yard crane at home, and only leave home for show-off purposes (i.e., you won't be using it to drag around other trucks on the road), then the Historical Vehicle plates should work well for you. If you plan to use it as a functional wrecker on the road, then you might need to pay the huge fees for regular commercial plates.
In either case, you will need at least a class B commercial driver's license to drive it legally here. You should probably get the air brake endorsement, too, as the screwy air-over-hydraulic brake systems on may US military trucks are sort of in a gray area, leaving a cop or DMV dweeb lee-way to decide if they want to ruin your day or not.
