I would consider placing the new tank in different location if possible and leave fishes and inverts alone until the new tank has been set up and allowed to run for a month minimum.
Otherwise,I would get everything needed to set up new tank and then place fishes in large Rubbermaid tub filled half way with tank water once you are ready to drain the older tank,and move filter and heater to the Rubbermaid tub while you set up the new tank which should not take more than a couple hours IF everything is ready to go.
Can remove plants from old tank if you are planning on using them in new tank and place them in another small tub.
The tubs are cheap at Walmart.
Once the new tank is set,planted,can move the fish/heater to the new tank, and place the filter on the tank and let it run with new filtration for a few weeks.
Monitor the water for ammonia and or nitrites daily and perform water changes if dictated.
Would it help jump-start the temporary tank by re-using the water from the old tank? When you remove the water so you can move the tank, can't you just put it into the temporary tank with the old filter. that way, you'd have the cycled water AND filter?
No benefit to using old water in new tank other than maybe 50/50 mix to help prevent osmotic shock.
Good or OK to hold fishes in temporary tank with old tank water while you set the new tank for same reason.Move fishes before you pull out plants to prevent ammonia spike from disturbed organic matter in substrate.
If water changes have been consistent,regular,then water will not be much different than clean water from the tap and clean water is in my humble opinion more beneficial to fishes than old tank water.