It is tough to get rid of nandina. Nandina is a hearty, well-adapted plant. I would bet that this drought has not affected it much. That is one of the many advantages of this trouble-free shrub.
If you must get rid of it, the methods you described will do just fine. Leave the roots where they are; they will improve the soil as they decay.
As for the nearby trees, if you pile a bunch of compost and mulch over an area where tree roots are, you will stress the tree. It just depends on how much compost and mulch you are considering and how close it is to the tree. Also, consider the mulch. I recommend using coarse mulch as the larger chunks of wood will allow for aeration. Look at the drip line of the tree. From what I understand, if a bed is built on or within the drip line, the tree roots will not get the air they need and the tree will suffer.