First off, I'm skeptical about any product that encourages you to breathe through your mouth, particularly for fighters. Secondly, what you are feeling is not a lack of cardio per se'. What you are experiencing (assuming you meant your FIRST BJJ class) is the cumulative effects of: A) Being very tense, as all newer grapplers are because you are unsure of yourself and you want to "do well". B) lacking in technique (if you don't have a move from a certain position your only choice is to muscle it and this takes waaaaay more energy than executing a technique) and C) not knowing how and when to breathe. (the tendency is to hold your breath when really exerting yourself. It may be only for a fraction of a second at a time but this is detrimental to your gas tank, think holding your breath on a treadmill....no good. And even senior students catch themselves doing it from time to time so...)
Once you have go to moves from each position and you learn to relax and breath and only explode for a second or two at a time when you need to you'll find you have waay more "cardio" than you thought.