I think that if you want to remove the character stats completely, you should instead give the cards the stats, and the character a stat modifier, with both characters' HP remaining the same (i.e. out of 1000)
For example, if Mr. Strong equips an armor card that gives 50 defense, he will increase it by 25% to 62.5 defense.
If you want to add strategy, you could give cards the ability to change your own or your enemy's modifier. If you're gonna do a one on one battle, though, it should be in real time.
Hmm, that's actually good idea I think and could be developed.
I don't agree real time in one on one battles though I see where you are coming. There is many games with one on one battles, like Pokemon and Kongai (though there is still switchable characters) and at least to me (and supposedly to many others) Deckmake was rather addicting even though it's battle system was so simple. So I don't think it would be a problem if designed correctly. I know that if I should implement multiple character fights, the project wouldn't be finished.