Not sure what you mean about "half-cycled water". Who cares about the water? It's the filter that matters, and there's virtually no filter bacteria in the water. So take 25% of the filter media from the established tank and stick in the filter in the hospital tank. I find air-powered box filters are excellent for this, because you can dump in ceramic, filter wool or whatever your mature filter uses.
Anyway, saltwater dips don't do any harm to the fish. The basic protocol is this: Make up a litre or two of full-strength seawater using some sort of salt. Marine salt mix is the ideal perhaps, but rock salt or tonic salt will do fine. Normal seawater is 35 grammes of salt per litre. Add the salt to water taken from the aquarium, so there's no temperature shock. Dip the fish into this for anything up to 20 minutes. The cut-off point is where the fish rolls over; that implies you've left the fish in too long. So dip the fish for as long as it looks fine. Archers are extremely salt tolerant, so you should be fine dipping for up to 20 minutes.
Repeat daily. I usually find that after even a couple of days the outward signs of the infection will have died right back. The antibacterial or antibiotic will then be able to clear up the actual infection.
Obviously, this is from my own personal experience, and your own experiences may be different. If you aren't certain, do short dips (5 minutes) to begin with, and see how things go. But I will tell you that while fixing slime disease on two dwarf red-tail puffers I dipped them for 20 minutes and they were fine. Like archers, puffers are a salt-tolerant group.
Cheers, Neale