You will not be able to root the entire (whole) thick branch.... do take tip cuttings of firm new growth. at lest 3 nodes and 4 inckes long apply 1-2 % IBA (rooting hormone) stick in well drained soil & mist taking 6-8 weeks to root.

Did you check for any life anywhere in the bark up and down the tree? You can use your fingernail and check for any green, but actually with Japanese maple you can see if the tree is alive as it goes from green to brownish-red where it is dead. If it has any life, you might be able to save it but may need to cut it back quite a bit and protect from direct sun. We had two in a pot on the deck that burned and leafed out again when moved to our garage to recooperate. Good luck, let us know what you find out.

First, mole are not the problem - they are carnivores and don't eat plants. There is a possibility of some other sort creature that might have fed on the roots (voles?) but can't confirm with the photos provided. Regardless, the tree now has a very compromised root system and will need some serious TLC if to survive. You have two types of JM showing based on both red and green branches. Green is most likely the root stock (straight Acer palmatum) and the red is the grafted top portion, which I assume to be a red laceleaf by the shape/habit of the tree from the first photo. So no need to graft any of the green further down - that's already what it is and will very likely sprout some latent shoots all on its own. All I would suggest for the time being is keep it out of direct sunlight, water as needed and just keep a close eye on it. No more fertilizing!! One should never fertilize a stressed tree and it is doubtful the Happy Frog is of much value in a container situation anyway (dry organic ferts are not very effective in a container - only with inground plantings). You can use any water soluble or liquid fertilizer you prefer but wait until at least midseason when the tree has stabilized a bit from the transplant shock and root disruption.

how about a pic ... dont be surprised if it drops all the leaves.. if it doesnt.. you are winning ... no sun until it grows some new roots ... did you properly prune any broken roots ... the key here.. is you are trying.. win or lose.. you will learn something ... so relish the challenge.. but dont set your hopes too high ... below is a general planting guide for trees .. if its small enough.. a pot might be best until fall ... ken https://sites.google.com/site/tnarboretum/Home/planting-a-tree-or-shrub