I would prune away everything that is dry and brittle but leave branches that are still green inside even if they are not leafing out

Also slide your finger down the trunk until you touch the top of the first roots exiting the trunk.Pull back the mulch and soil to this level 3or4" round the trunk to let it breath,this is the correct planting depth.You can get away leaving it like this as long as water doesn't pool in the SHALLOW bowl you may have created but it's best to raise the plant up. If the first roots are more than a couple of inches below the soil then I'd raise the plant up to be safe.Now is an ideal time,you could dig out the whole rootball untouched...don't fertilise until established.

I am sure you realize that once you prune out the dead wood you're not going to have much tree left :-( The dead wood is fairly obvious even in the photos - it is very gray compared to viable wood.

The point I'm trying to make is that once pruned, the little tree is going to look kinda butchered. It may take a good few years before it starts to approach anything like a 'normal' appearance. I understand the desire to save it, but if in a conspicuous place, you may want to consider moving it to a protected, out of the way location and replace it with something else.

No fertilizer at this point. Just prune out the dead and hope for the best.