I think the best way to handle this depends on the reasons for breaking into your room. When I was a preteen/teenager, I frequently went through my mom's stuff, drawers, cabinets, whatever. I can't even tell you why I did it, I wasn't looking for anything specific. I think teenagers are just curious what kind of things their parents have in there. Its like finding clues to the kind of person your parent is. After I was grown and having children of my own, my mom told me things about her life that I never knew, so obviously she was able to hide the most embarassing/private issues from my sneaky snooping.
I agree with the posters suggesting a locked safe or other form of unpickable locking box for the dangerous meds. However, to discourage attempts at break-ins solely out of curiosity, I would open the safe and distribute meds right in front of her, even shuffling around whatever else is in there, so she's aware of everything in the box and why you lock it up. (If suicide is really an issue, you might consider locking other dangers - razors, etc. - in a safe as well.) I also agree with keeping the bedroom open if only the meds are an issue. And don't make a big deal out of curious snooping as long as you know she's safe.
I don't know much about bipolar disorder, so I can't make any suggestions based on the mentality associated with it. But I know that the feeling of being trusted is very important to any teenager. I think it would help your dd to know that you take precautions not out of mistrust for her, but out of the risks of her condition. Let her know that everyone makes mistakes and bad decisions. Point out some of your own. Give examples of families in different situations and the precautions they must take. (Parents of children in wheelchairs put guards on stairways, not because they think their kid is too dumb to avoid the stairs, but because accidents DO happen.)
Good luck to you, I hope you find a solution that works well for your family.