Probably possible, but difficult, based on my experience.
When I was committed after my detected, failed attempt to CtB, I ended up in one of the "nicer", privately run facilities in the US, (a dedicated floor in a hospital) and considered a high risk for finishing the job. My room was equipped with a camera and some low intensity red light at night so they could monitor me (think like a dark room... effing terrible, having the room bathed in blood-red light at night), and of course they took away the obvious ligatures like my belt and shoelaces.
Other people who weren't considered as high risk were put in rooms without the camera and blood-red light. Not super secure, either way. All the rooms had doors (which closed, but didn't lock), those standard, movable hospital beds with locking wheels that could be used to block the door, ceramic sinks and glass mirrors that could be shattered for a slicing implement.
During the daytime, all us unsuccessful suicide cases had the run of the floor and the staff kept an eye on us, and at night, they were *supposed* to do rounds every 15 minutes, but I clocked them, and they slacked off a bit, especially around shift change and late at night once they thought every one was sleeping. During their rounds, they made sure the doors to the room were open and just did a quick look from the hallway. Guess they would have seen anyone obviously bleeding out or obviously hanging themselves, but I doubt they would have noticed a ligature compression suicide under the sheets in time (and they didn't take our clothes, socks or underwear).
So, yeah, I saw a couple ways to go inside a psych ward, but nothing I can recommend. Most likely outcome seems to be that you'd be "saved" and in a much worse way than before - even if you don't end up brain damaged or otherwise seriously injured, an unsuccessful suicide attempt inside would definitely lead to a longer stay. I'll admit I was very, very tempted to try anyway (and I was only in a few days), but not a good idea.