Mdthbs

In Curse of Strahd, my cleric cast Banishment on the (possible spoiler):

Shadow Dragon

At the end of Banishment's duration, will the creature return? I don't know what plane it's native to. Our DM seemed unsure, himself. Our party has reason to believe that this creature is:

a silver dragon that was turned in to a shadow dragon by Strahd.

I don't have any confirmation, but I believe that the creature actually is:

Argynvost

In Barovia, the spell fails unless a specific condition is met

All that being said, in Barovia (emphasis mine):

Astral projection, teleport, plane shift, and similar spells cast for the purpose of leaving Barovia simply fail, as do effects that banish a creature to another plane of existence.

... so the banishment spell has no effect until:

Strahd is defeated, [when] the fog that surrounds Barovia fades away, allowing the inhabitants of the valley to leave if they wish.

It is unclear if that event removes the restriction to planar travel or only permits inhabitants to leave through that specific method.

It's unclear for shadow dragons in general

Shadow dragons, according to their Monster Manual entry are:

true dragons that were either born in the Shadowfell or transformed by years spent within its dismal confines

In the former case, their plane of origin would be the Shadowfell, in which case, they would not return, remaining on the Shadowfell. Along a similar vein, a dragon born in most other planes who then was transformed, would also be permanently banished to that different plane.

On the other hand, a dragon born in Barovia who was transformed from some other effect would return (after being banished to a harmless demiplane).

The shadow dragon in Curse of Strahd:

I could not find a reference to one, so it is possible your GM added it to the campaign. If that is the case, it is up to the GM where this specific shadow dragon originated from.

If it is, as you suspect:

Argynvost

... then the dragon's origin plane is still unclear, as no mention of it is found in the module except that:

he had come to the valley years earlier in the guise of a nobleman named Lord Argynvost.

Without any further details of his origin, it will be up to the GM to decide if the dragon is native to Barovia or a different plane.

The spell fails and the target remains.

Page 24 of Curse of Strahd states:

Astral projection, teleport, plane shift, and similar spells cast for the purpose of leaving Barovia simply fail, as do effects that banish a creature to another plane of existence. These restrictions apply to magic items and artifacts that have properties that transport or banish creatures to other planes.

When the Cleric casts Banishment, the spell will fail as a result of this alteration to magic.

Shadow Dragons are said to be from the Plane of Shadow

From this entry on Forgotten Realms Wiki

These dragons from the Plane of Shadow were seen in the Frost Hills and the Thunder Peaks.[3] They could also be found in the Dragon Eyrie.[17]

So the Shadow Dragons are considered to be from another plane of existence and due to that, it is the following part of the Banishment spell's description that should be used:

If the target is native to a different plane of existence that the one you're on, the target is banished with a faint popping noise, returning to its home plane. If the spell ends before 1 minute has passed, the target reappears in the space it left or in the nearest unoccupied space if that space is occupied. Otherwise, the target doesn't return.