Shadows are fairly docile if not attacked (indeed, if Dante stands perfectly still they will often just circle around him, not doing anything), so it is best to focus on taking the creatures on one at a time while listening out for the sound cues of any attacks the others might be performing. The ideal tactic is to go for critical hits: these can be performed by waiting for the Shadow's Skewer attack, then jumping on the spear while it is extended and firing Devil Trigger Ebony & Ivory shots at the Shadow. Success will result in hits dealing damage directly to the Shadow's core, which can kill it in a single volley on most difficulties. Shadows always drop health when defeated, usually enough to completely refill Dante's health.
If the player puts off doing this secret mission until New Game+, Nightmare-β is ideal here as it can very rapidly put Shadows into their vulnerable core state.
If the Shadows defeat Dante, after the use of a Yellow Orb Dante will be returned to the workshop and the dial on the wall can no longer be activated: this is what the mission description means by having one chance. The only way to try again is to reset and start the mission over. Saving before resetting saves the mission as having been failed, and will render it inaccessible until New Game+ if the player does not have a prior save file to revert to.