It's not clear that the ring would even activate
The operative part of the Ring of Mind Shielding's description is:
If you die while wearing the ring, your soul enters it, unless it already houses a soul.
However, the language used to describe the lich's abililty to reconstitute itself from its phylactery seems to carefully avoid the words "death", "...
It would seem that both RAW and RAI is that it would summon another avatar
The text of the card's description says (emphasis mine):
It appears in a space of the GM's choice within 10 feet of you and attacks you, warning all others that you must win the battle alone.
Lets look at RAI first. It would seem that the PC has to defeat the avatar on their own, ...
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 ...
There's a game mechanic that provides some chance of getting the soul out of the ring, but not by forcing it.
A way according to RAW for the soul to leave the ring is if it chooses to.
The ring's description states that the soul inside it
can remain in the ring or depart for the afterlife.
Therefore you could attempt a Persuasion check or a Deception ...
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 ...
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.
All Spellcasters (with limitations) and possibly Thief Rogue
It is written in Spell Scroll description
If the spell is on your class’s spell list, you can read the scroll and cast its spell without providing any material components. Otherwise, the scroll is unintelligible.
While you can't cast spells from other class lists it still provides versatility, ...
The lich is not forced to stay in the ring
In the description:
If you die while wearing the ring, your soul enters it, unless it already houses a soul. You can remain in the ring or depart for the afterlife.
The key is that the lich CAN remain in the ring. But it can willfully leave to the afterlife, and then to their phylactery whenever it wants.