I have played music for most of my life, and like many kids started with band. I've found over the years the best place to go is Craigslist. Yeah, it's used, but it's the best deal around. I've bought and sold so much stuff off craigslist, and as long as you ask the right questions and check it out before buying you can't go wrong. They have some definite bargains. Buy one used, clean it up before you give it your kid, and they'll never know the difference - and have the benefit of having a better instrument. It's true that the better an instrument is the easier it is to learn. I had a crappy guitar when I was 12 or so and barely played the thing because it sucked. Once I upgraded I couldn't put it down. Quality makes a big difference. At least consider Craigslist an option. Just go to their musical instrument section and search for clarinets.
For example here's the clarinet that someone referenced that goes new for $900, selling at $425.