The Twinkle variations (Ah vous je dirais) aren't really that bad, except for the last variation. However, even with your practice time, I think you can be smart about it.
Practice the most difficult parts ONLY. Do not waste time playing the things you can already play. Get the hard parts up to the same level of ease as the easy parts. Play through the entire piece once every couple of days until it's all worked out. Then work with a metronome to gradually increase the tempo to where you need it (playing straight through the pieces now).
Also, you may want to try getting up earlier in the morning or staying up a little bit later - even an extra 15 minutes can make a difference here.
When at your lessons, ask your teacher for help in specifics on how to work out the issues you are having. For example, your LH being too loud, what can you do to make that softer? I have some suggestions, but it's much better if you go to your teacher for this.
I think this is a great opportunity for you. It's not a competition where there's a winner or loser, right? It's more just a personal goal to help jump start things a bit. I'm guessing your teacher sees a lot of potential in you, but it's unrealized because you're not as serious as you could be, and this is one way of getting you there. When you really stretch yourself, you can make great strides.
On a side note, not being able to play a scale doesn't necessarily mean you aren't good enough to play. You just probably haven't practiced scales enough to be good at them. It is worthwhile, however to be able to do and will help your Mozart pieces as there are scales all over the place. Use a scale as a warm-up and play through them hands separately at least once every day. If you mess up, start over and go slower until you can play it - no matter how slow. Then when separate is easy (however many weeks that takes), put hands together. You'll get there.
edited to add: well, this is embarrassing. I didn't realize this was an old thread. I'll leave the above up for some other poor soul in a similar situation though. Congrats on your results