Do you have one of those "breeder nets" you can place him in? - they aren't ideal for cories as they must remain at the upper level of the tank but cory can rest on the fine mesh at least & just focus on breathing, just place the net so that it receives steady, gentle flow.
Do check back in with the shop as it's unlikely to have happened just in your tank, perhaps they've been treating these fish & will have some suggestions.
In all honesty, once a fish is behaving this weak, it's unlikely to recover so isolation from your system is best ... more stressful for the fish however (which can be a death sentence right there).
Increased slime production (that greyish film) is a basic fish defence that can be triggered by many causes, external parasites are always a good choice in first treatment as these opportunistic little beasties are always present in small numbers (as are most representatives of the germ microcosm) & can explode in population v.e.r.y quickly ... but they also tend to respond to meds which are very effective & well tolerated by fishes, even sick little dudes.
As always
1) water change - do this daily for the next 1-2 weeks if possible
2) optimize oxygenation of water column - surface ripple, gentle splashing, micro-bubble air "stones" that also improve surface movement, possibly the Twinstar type device (not sure how frequently one can set these to "run" if one has the "disease" version rather than the algae version)
3) treat for external parasites
I use formalin - not sure how available this is for you (chemist, vet, fish shop???) - as it's easy to remove from system & remains one of the most effective treatments for external parasites; most shops will try to sell you a malachite green or malachite green/formalin formulation, the latter is much more effective than just mg, acryflavine based meds are slower to be effective (& are linked to longterm internal damage to organs such as kidney/liver/reproductive system) so I leave these for later use in any treatment "trials"
Unless the particular "nasty" has been identified (microscopically, eg skin scraping, gill scraping, etc - note that the list will always come back as a "list" with hopefully just a few dominating species that are well described), medicating fishes is guesswork.
Unfortunately the "solution" that many companies take is to compound formulations, mix in some compound with anti-parasite activity, then add in some anti-bacterial, then some anti-fungal ... in the end, very few of these supply therapeutic levels of the assorted compounds (sometimes it's a regulation issue, sometimes cost, sometimes ignorance)
Be wary when treating corydoras, as they can be sensitive to many "treating" compounds, healthy cory's may just look lethargic & unhappy (I swear they glare at me) in response to meds, but the sick fellow you're trying to save, may just keel over.
Some will suggest adding salt for external parasites (anything really that triggers slime coat) but if you actually look at the scientific data indicating salt levels that are effective against various parasites vs salt levels that many fish prefer, it's not the most logical recommendation (ie, add a teaspoon/7.5 - 35 litres if it makes you feel better, won't hassle the fish too much
- with cory's always begin conservative, increase salt levels every 12-24 hours BUT more importantly, do those water changes & then the salt maths)
Basically salt works by irritating fish membranes so that "slime coat" is produced - not just the visible skin sort but also increased secretions across the gills.
I could go on (& on) but will stop here.
You might post specific questions (which I might be able to answer)
