Once the new top growth is present timing for digging up is terrible, especially on deciduous kinds, which may shrivel if dug when leaves are young and soft.

Do not dig compost into ground for shrubs unless entire potential root growth zone for many years is amended. Much better just to plant in (loosened) existing soil, use compost as mulch. Planting long-lived plants in small holes, pockets or zones of amended soil places these at a disadvantage, due to how water moves into, through and out of different soil textures.

"Absolutely too late"? I'm not sure there is such a time :-) If care is taken to dig and maintain a sufficiently large rootball, one can transplant at any time in this area.

Yes, there are more ideal times than well into the growing season but if circumstances dictate, transplanting can be done any time. Be sure the plant and surrounding soil has been thoroughly watered before attempting the move, pick a day with cool cloudy weather if possible or dig late in the day, have your new planting location already prepared and replant ASAP. Water in well.

Transplanting is not all that different from new planting, except that you are dealing with an established root system. If you can reduce or eliminate disturbance to the bulk of the root system as best you can and minimize transplant shock, you should be fine.

Cutting the roots of a deciduous shrub in June is quite likely to result in collapsing of the foliage. Specimen will probably not die, if kept well watered, but will look bad for some time.