Sharpie works and is fast. Yes it will fade eventually, not before OP has a chance to unload and organize in new home. Fade will be less so from the ingots in the middle of the pack not exposed if you mark the sides. Harder to read on a shelf than marking the ends.
Harbor Freight stamp set is certainly most effective long term however it stinks for providing detailed information. Works fine to L2 on some Lyman #2 alloy or WW or Pb for clip on wheel weights or plain. Not so good for 2.5/4/93.5 alloy or 6/2/92 alloy.
BTW - I always put Sn first, Sb second. Do the math to figure out the Pb remainder. So that 6/2 alloy is 6 Sn, 2 Sb casts great not super hard but hard enough for pistol and will spread on impact.
My advice for the move is use the sharpie to mark what it is. Stamping it all is a nice little cold weather project when you are not trying to pack for a move. Got enough tasks on your plate I'm sure.
Container will depend on ingot shapes and size. I like 5 gallon buckets. You don't have to cut wood round, it will be stronger if you do have it fit the bucket bottom but a square piece you can rip with a circular saw from a sheet of plywood, OSB or 3/4 board will prevent piercing the bottom of the bucket and should be sufficient. I find one can stack 50# of COWW's in 5 gallon buckets 3 or even 4 high without the wood but they don't have corners likely to pierce and are not going for a ride. The 4 high lift is a bit of work though. For storage right by the smelting I have gone 78# per bucket and 3 high.
If your ingots fit a USPS Med Flat Rate Box they are sturdy enough if well taped to handle 50-60 lbs. of lead. Easy to label too.
Any way you slice it 1,200 pounds in 50 lb. packages is 24 containers. Best if they be something you can move with a hand dolly carrying them all would be a mite of work.