I have done this move a few times with my 120g. Most recently 1.5 years ago. I have also helped many local reefers move. You want to make up as much new water as you can before the move. I mix the water in as big a container as you can find but for the actual move I use 5g buckets.
Do NOT use any of the old water once you start disturbing the sand bed even a little. And definitely give that sand a very very good rinsing before you put it in the new tank.
40g is such an easy move. I would get 12x 5g buckets. I prefer the old salt buckets as they have the rubber o-ring so no water drips out in transport. Your pickup bed will be fine for the standard home depot buckets and lids though. Just give it a good wash down after if you don't have a liner.
The coral frags I can pick off the sand bed or easily pull off the rock gets bagged or in tupperware. The rest stays on the rocks and goes in the buckets.
For any tank move I usually calculate 75% of total water volume including sump and mix up that much new water beforehand. I do it a day before and have powerheads pointed at the surface the whole time to make sure it is well oxygenated.
Day of the move I would fil 10 of your 5g buckets with 3g of the new water. I then fill some tupperware with old tank water for the frag transport.
I then pull as many corals as I can off and get them packed away.
Next step is to top off the remaining 10 buckets with 1 to 1.5g of old tank water that should still be almost crystal clear.
I then fill the extra 2 buckets with old tank water. We don't care if this water is clear or not as we use it to rinse the rock when it comes out before we put it in transport buckets.
Now you are ready for the dirty work. Pull the rock out with what ever coral is attached and give it a good swishing in one of the buckets full of old tank water. You will be amazed at the amount of waste that is stored in the crevices. Give it a second rinse in the other bucket of old tank water. Then place in one of the 10 buckets with new salt water.
Repeat this until you are down to just sand.
Empty the 2x 5g buckets that you just used to rinse the rocks and fill them with sand. Do NOT put this sand back in your tank without thoroughly rinsing until the water is totally clear. There is so much waste in that sand it can cause serious issues. There are many threads on here about tank crashes after disturbing sand bed. Most people use a garden hose ans rinse the sand with fresh water as the nitrifying bacteria housed in a sand bed is very minimal. I always keep a cup or two of old sand and rinse it with salt water in an attempt to keep the biodiversity alive but this is probably overkill probably pointless as the sand bed will surely repopulate fairly quickly with what is on the live rock. But i do it anyway because I'm crazy. I do this rinsing long after the tank has been moved and setup. I usually run the new tank a couple days bare bottom as the sand rinsing is the biggest pain and i always lack the motivation to do it after a big move.
Once you have the new water mixed up its really only a 2 hour process to break that tank down and get things packaged for the move.
I always make sure I have a couple days where I'm able to use both places but if you cant do that then the tank break down should be the last thing you do before vacating the old place and the first thing you setup at the new place to minimize the time the stuff sits in those buckets without oxygen. Longest I have let them sit in buckets is about 12 hours but I'm sure they can go longer.
Have fun.
Here is a pic of the 120g i have moved a few times. Maybe 50-60 pieces of coral and more lbs of rock than I care to think about. Never lost a piece of coral and only 1 fish died from being squished by a rock.
