How do Dimensional Shackles interact with an Arcane Archer fighter's Banishing Arrow?

Question

An Arcane Archer fighter hits a creature with a Banishing Arrow, and they vanish into the Feywild.

A man in the middle places dimensional shackles on the target (in the Feywild) for 10 minutes.

What happens when the shackles are taken off?

Answer 1

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 Arrow's effect, nothing else is in place to remove it from the Feywild.

Answer 2

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 movement, including teleportation or travel to a different plane of existence.

Even though the shackles specifically state that:

They don't prevent the creature from passing through an interdimensional portal.

Interpreting the banishing arrow to be considered an interdimensional portal would be an even bigger stretch than it being considered a type of movement between the planes.

However, the banishing arrow specifically also states that:

You use abjuration magic to try to temporarily banish your target to a harmless location in the Feywild.

So whether or not a location where some other creature puts on the shackles would be considered to be a "harmless" location is also something to consider.