If we search for the word "stage" in the novel (Theatre) using the PDF document linked in the question, there are 70 instances. The name Hamlet is mentioned 8 times:
- seated on a gilt Hamlet stool
- Over the chimney-piece was an old copy of Lawrence's portrait of Kemble as Hamlet
- "I wonder if I'm too old to play Hamlet. Siddons and Sarah Bernhardt played him.”
- "Don't be idiotic, Charles. I wouldn't play the Queen. I'd play Hamlet." etc.
I haven't read the story but it appears to be centred around an actress whose ambition is to play the role of Hamlet on stage. It's my assumption that the gilded stool refers to a rather expensive-looking stool the theatre used as a stage prop in the play Hamlet. In this instance the "Hamlet" is a (proper) noun adjunct acting as an adjective.
On Google images I searched for "19th century gilt stool" as the story was first published in 1937. It is possible that the stool in question resembled the one below; it is British-made and certainly looks theatrical.
The renowned auction house, Christie's, describes the following piece of furniture:
an English green-painted and parcel-gilt stool [partially gilded stool] possibly early 19th century, after a design by Thomas Hope