One of my dogs had bad separation anxiety. There are actually a lot of websites that discuss this issue. Some say to leave your dog alone for short periods at a time, when she starts whining, to go back in and tell her no and then continue this process slowly for longer periods. Long story short, this didn't work with our dog.
Also, if you give it too much freedom initially, it causes problems. Some dogs take to the kennel, others don't. Ours hated the kennel with a passion.
In the end, we did kind of end her separation anxiety towards us by getting another dog. Now we have two and they keep each other company when we're not around. In fact, sometimes, they'll just saunter on to another room to snuggle if we silly humans are keeping them awake or something. Personally, I think dealing with separation anxiety is a trial and error sort of thing.
Good luck! It's hard. Another thing I would strongly recommend for anyone with a dog is to take it to obedience training. This also helps bonding with your puppy, socializing it with other dogs and also establish with the puppy that you are the alpha. Bichons typically have good dispositions, but it's so much fun and you get to meet other puppy parents, plus the trainer can give you lots of tips on raising your new family member. Good luck!!