Unfortunately, the way hawlucha sleep in the wild is actually the ideal position for a trained hawlucha to sleep, so changing that isn’t particularly something I would recommend.
However, there are other things you can do to keep your hawlucha from experiencing back pains. Giving him a massage after every battle (or daily) as well as a warm compress will help his muscles to relax and, therefore, heal themselves. Additionally, some fighting-type pokémon—even in the wild—engage in their own forms of meditation or relaxation techniques to avoid the exact problem you’re describing (or muscle issues in general). Consider engaging in guided meditation with your hawlucha to ease his tension further.
Of course, if you haven’t already done so, adding exercises to increase flexibility and core strength may also help by stretching the muscles and reinforcing the abdominal muscle. The latter is handy by taking pressure and weight off the back muscles whenever your hawlucha performs day-to-day activities. Not to mention it strengthens his front so that moves such as Flying Press are much less of a shock to his system.
And finally, most importantly, ensure he’s standing straight when not in battle. If he slouches to eat, while walking—anything, this could further strain his back.
In short, while correcting one’s sleeping position is the easiest way to alleviate back issues in humans, in bird pokémon, it’s far more complicated. Instead, it’s far better simply to treat or prevent back pain in a bird pokémon’s waking state, when it’s active and can adopt changes to its routine.