The ones we can place right away are the Shadow and the Peasant, at 1 and 12 respectively. To be trapped between a shadowy sword and the Knight's sword, the Dragon needs to be at 2 and the Knight at 3. That automatically places the Merchant an 8. The Gypsy should be right next to the Merchant, and the Cock right next to her, so the Cock is either 10 or 6. But the place right across the Cock should be empty to fit the Hydra, so it has to be placed at 10, which places the Gypsy at 9 and the Hydra at 4.
For now, the only unoccupied places are 5, 6, 7, and 11. 5 and 11 have to belong to the Castle and the Giant, as they are opposite to each other. It is impossible to determine which is which, but this is not relevant to the solution. That leaves only 6 or 7, and we are out of characters.
The characters are capitalized in the verses, so it might be the one that coincides with the first word that one of the verses starts with. In that case, only the Wagon fits. For some reason, the word 'crouches' starts with a capital letter, too.
If the Wagon is one of the characters, that places it next to the Merchant as 8 and the only place left is 6, which becomes the solution.