We have a very similar question about a ring of mind shielding, and the same reasoning applies here. Liches do not die when their bodies are destroyed, so the blackstaff does not trigger. Also, a lich's soul is already stored in the phylactery (and not in its body), so stuff that tries to take a lich's soul from its body is unlikely to work.
I would say that if interpreting the banishing arrow as a means of movement between the planes (in this case both forward and back), the Dimensional Shackles would prevent the secondary trip back to the plane of origin.
In addition to serving as mundane manacles, the shackles prevent a creature bound by them from using any method of extradimensional ...
The creature remains in the Feywild
According to Dimensional Shackles:
the shackles prevent a creature bound by them from using any method of extradimensional Movement
The only "return clause" in Banishing Arrow says
At the end of its next turn, the target reappears in the space it vacated
If its turn passes and its unable to fulfill Banishing ...