Jada . . . hmm. Yes, that sounds about right for treatment. If we're thinking that his lethargy is from an internal infection then the fastest way to get meds to that internal infection is via medicated food. But Furan-2 isn't used for medicated food; Kanaplex is.
So, picking the right antibiotic depends on what you want it to treat. If it's external, Furan-2 will work. If it's internal, go with the Kanaplex. If we're not sure, go with the Kanaplex since it will also absorb into the gills whereas Furan-2 won't.
o.O Goodnes, it really is confusing, isn't it?
Now, the General Cure. Did you just start dosing it tonight? And what size tank? A convenient way to dose General Cure is to get a container that holds 5 cups of dechlorinated water. Add the dosage of General Cure and mix well. Then add 1/2 cup of medicated water per 1 gallon of tank water. So if his tank is say 3 gallons, you'd add 1.5 cups of medicated water. You can store the rest of the medicated water for a day or two and reuse.
It is true many meds do reduce the oxygen because that's how they can kill aerobic bacteria; aerobic bacteria uses a lot of oxygen to thrive. For bettas, who can breathe supplemental air from the surface, it's not quite so dire to have aeration in the tank but it never hurts. :)
So, since I'm still confused by my own post, let me sum up:
if you have it, use the Kanaplex for antibiotics. If you are mainly trying to treat an internal infection, medicate food with it. If not, use it in the tank with the General Cure.
In a case like this, do not stop with the Kanaplex until you've done at least two courses (6 total doses). I don't want to see any bacteria come back resistant to meds.