The Rev Dodgson said:

Bubblecar said:

Bubblecar said:

FAKE

I’m now thinking it could also be SCAM

The word must have at least one unique letter and at least one letter that only appears in two words. The possible words are:

BAIT, RIGS, FAKE, SCAM.

If she got a T she would know it was BAIT.
If she got an I Doug would get an A or an S and not know the word.
If she got an A Doug would get an I, T, or S and know the word, so she still wouldn’t.
If she got an S Doug would get an R, I, or A. He wouldn’t know the word if it was A or I, so he must have got an R, so the word must be

RIGS.

If Amy has B, Bob has T, Carol has A and Doug has I, Doug wouldn’t know whether it was BAIT or RIGS.

If Amy has G, Bob has R, Carol has S and Doug has I, Doug wouldn’t know whether it was BAIT or RIGS.