I've had a ton of trouble getting moderate care characters on the TF. I thought it was impossible until I finally managed to get a Tacttchi! I didn't max the weight out, though I didn't play any games and only fed snacks for the happiness levels. Honestly, I ignored it as often as possible, let it call for my attention all day, and I'm surprised I got a "Good" care character instead of a "Bad" care. My only advice is to ignore it as much as you can, get as many care misses as you can, and you'll get either moderate care or bad care. (Like I said, I thought I was getting "bad" but ended up with "good"/moderate somehow.)