The context is that the narrator is selling stakes “in three of the finest merchantmen beneath sail” on a “trading floor”, so “paddle” would seem to mean a bidding paddle of the kind used in auction rooms.
But “calendula” is a mystery. The only plausible explanation I can come up with is that “calendula”, the generic name of the marigold, is intended to refer to the colour of the paddle:
marigold, adj. Of the colour of the marigold flower, bright orange-yellow.
Oxford English Dictionary
This is plausible because the trading floor is operated by “House Gold”, which seems to be a fantasy equivalent of a Renaissance banking family like the House of Medici. Such a family might well decorate their equipment in their house livery, which would likely be a colour close to gold, to suggest their name.
If this theory is right, it would be a misuse of the word, because although “marigold” refers to the colour as well as the flower, “calendula” does not: it refers only to the flower. But since The Liar’s Key is a fantasy novel, perhaps the misuse is deliberate, to give a sense of estrangement.