Fin rot, kanamycin doesn't help

[nonphotosynt]

Hello, need help:
New baby fish developed fin rot in 2 days. Fast progress, in 3 hours was visible difference.
Both babies of the same kind - tassled filefish - were affected, while other same day purchase (lionfish) doesn't show signs of disease.
Started treatment by KanaPlex (Seachem's kanamycin). Dose should be given once in 2 days, maximum 3 doses. It significantly slowed rot development in the first day, but on the 3rd day of kanamycin rot accelerated again.
Here is photos at the moment. Dorsal and lateral fins are also affected. Pardon the quality, this is point and shoot camera and fish is constantly changing position.

What can be added during dosing kanamycin (3 more days)? What next?
If possible, not killing biofiltration in one day. I have cycled sponge and ceramic media in sump or another tank, but not enough to replace all biomedia (LR, actually) every day.
My possible course of action?

 

[sepulatian]

Are these fish in the DT or QT? Maracyn 2 for saltwater fish works well on fin rot. Are the fish eating? If so, dose all of their foods with a rotation of Vitachem and Zoecon. It will boost their immune system and help them heal faster.

 

[beth]

That may be more lymphocyctis, then a bacterial infection. What are your water readings? Size tank? What are you feeding?

 

[nonphotosynt]

Yesterday evening - slightly better. I added Nitrofurazone to Kanamycine, and in a few hours the thick white spots disappeared. Will see, how it will be today.
I have Selcon and Reef Plus, will they be enough for soaking food?
Fish was bought this Saturday, Nov 29. Signs appeared in the morning Dec 1. Kanamycin dosing started the same day.
Tank is not quarantine, but is used as such. I recently disassembled 90g tank, overrun by aiptasia, for a cleaning, all inhabitants are in separate containers, fish - in 20g long tank with live rock.
20g long, each fish is less than 2" long. Tank was set few months ago, but its content (LR, water) were transferred from 90g tank, existed for more than 2 years.
After a month in this tank (pseudo-QT) all inhabitants will be moved to 90g.
Water parameters:
ammonia, nitrites, phosphates are zero, nitrates 5 ppm, alk 9 dKH, pH 8.2, 77F, SG 1.026.
Tankmates:
2 years old percula and blue-green chromis, both don't bother newcomers.
Same day purchase: less than 2" antennata lionfish.
Only tassled filefishes are affected. Appetite is not affected, the fish eat fine.
Bought Gel-Tek Neomycin and Ampicylin, just in case, tried to drop drops in the water, without fish - the size of drops is too big for these babies with small mouths.
Feeding: mysis, SFB Plankton, Angel Formula, red algae.