[sciurius]

Let's see if I can follow.

To get A from Bb, you transposed down by a minor second,

Yes, a minor second = one interval.

Applying the same transposition to D yields C#.

One interval lower than D is the note between C and D, which is Db or C#, depending on the scale.

Since we do not take key signatures into account, I'm inclined to call this Db. What musical rule says it should be called C# ?