Yes, sir, I agree with your viewpoint — it is always preferable to eliminate a problem such as this using biological control as opposed to chemical warfare, when feasible.
Your aquarium is large enough that I don’t think one anemone will present a problem for the pipefish. Your dragonface pipefish will be wild caught and savvy with regard to anemones and stinging corals. They should know enough to avoid the anemone, and in a 120-gallon aquarium, they should be able to stay out of its way without too much difficulty.
I don’t know about the citron gobies and if they will be helpful in controlling an infestation of red bugs on your live corals or not.
I have no experience keeping dragonface pipefish, and I don’t know if they do well in large groups or if they may become territorial toward other dragonface pipes when they are kept together in the same aquarium. I do know that their normal social unit in the wild is the mated pair, so I would suggest starting out with a pair of the pipes and seeing if they can make a dent in the red bug population.
I would also suggest that you consider asking the reef keepers at Reef Central if it is advisable to keep more than one pair of dragonface pipefish together in an aquarium the size of yours, and if so, how many of the pipes they would suggest for controlling red bugs in a reef tank like yours.
Best of luck eradicating the red bugs, sir!