It seems logical that putting the guns and ammo inside the safe would keep them... safe, BUT...
I don't recommend doing that for a couple reasons.
First it would weigh a ton (literally) and most moving companies charge extra for moving safes, often by the lb, so you would be paying for weight they would otherwise move at no extra charge with household goods. If the safe is going along with the rest of the furniture, the extra weight will be a problem in load balancing the truck, making the safe one of the primary considerations in load placement and other stuff will have to be packed around it, spacing for weight distribution- more work for the movers...
Then there is the issue of contents and damage- since the safe will likely be closed and locked, how are they going to insure it, and verify the contents, condition etc against any future claims at the destination- who's to say that scope wasn't broken before the trip began?. Where did that PM disappear to?
Then there is the legal issues of which I know nothing... however the case could be made that you are giving possession of firearms to the moving company... even if the safe is locked, (who may employ prohibited possessors) and they may need a FFL license to move them across state lines (or think they do) -which they probably don't have- so they don't want anything to do with guns. I don't know how you can not mention them if it obvious your moving a gun safe, but you'll think of something...
I just moved, and I sold my safe to the people who bought my house (so I didn't have to move it) for more than I paid and then bought a replacement in my new location. Let me tell you, moving guns and ammo is not easy. One 50cal can of ammo weighs about 30# -any quantity adds up quick... and depending on how many rifles/pistols you box up together, they get heavy also. It's better to separate things into smaller units that don't weigh as much. I moved the ammo myself in my pickup and didn't say anything about guns to the movers.
As a theft precaution, I removed all the bolts, slides and cylinders and packed them separately so that the gun was incomplete and not resell-able in case it was lost. (in fact, I keep most of them at home this way in case of a break-in... makes it less likely they will be taken if they don't have all the pieces). What I would do is pack a number of long guns individually and then group 2 or 3 together in other bigger boxes so they don't look like gun boxes. Separate and pack AR's uppers/lowers into different smaller boxes, pistol frames and the slides/cylinders/bolts into separate boxes.
I number all boxes on all 4 corners so they can be identified without opening (you have to make a list of contents as you pack). Each room or item category gets its own number series... 100,200,300... I give the guns & parts a distinctive number series so I can find them later. Anal I know, but it works for me....
good luck on your move.