Each of these clues yields a two-word phrase where one word is a constellation, and the other is an anagram of the constellation with one letter replaced. Reading the new letter in the non-constellation spells out BALANCE HAIRSTYLE, which clues the phrase LIBRA BRAID, which has the same anagram property. The non-constellation element, BRAID, is the answer.
Here are the answers to all the clues.
- broadcasted dolphin — PUBLISHED DELPHINUS
- no-nos for herdsman — BOOTES TABOOS
- swan, cozily — CYGNUS SNUGLY
- twin paradox — GEMINI ENIGMA
- wandering cup — ERRANT CRATER
- saves Medusa slayer — RESCUES PERSEUS
- resentments for small dog — CANISMINOR ACRIMONIES
- cherry for big dog — CANISMAJOR MARASCHINO
- AirTrain for small lion — LEOMINOR MONORAIL
- hero carver — HERCULES CHISELER
- more ersatz firebird — PHONIER PHOENIX
- red lizard — SCARLET LACERTA
- tempest for small bear — URSAMINOR RAINSTORM
- impudent fly — SAUCY MUSCA
- bedevils winged horse — PLAGUES PEGASUS
- smelly dolphinfish — ODORED DORADO