Yes, in the sense that due to the funding situation of such facilities a bunch of people get refused entry (or because of "complex" problems. E.g. suicidal + addicted? No help for you). So you can study the group that gets helped and the group that doesn't and compare them.

Of course the criticism is that they only take the worst problems (even though refusing "complex" cases obviously contradicts that more than a little bit in my opinion). Likewise other studies that assess the ability of professionals to predict who will commit suicide keep coming with very disappointing results, so frankly, I find it hard to believe they could select the serious cases even if they fully intend to do so.

But the group that gets help has a bigger suicide rate compared to the group that doesn't. Quite a bit bigger. So it depends what you believe.

1) do you believe the criticism that they are able to select serious cases (but are unable to prove that in studies specifically checking that) ? Then it is unknown how effective their treatments are, and until they quantise "seriousness" it is impossible to check how effective they are.

2) do you believe the selection process is mostly random and the criticism is invalid? Then the treatments are counterproductive and they cause suicide. That does not (necessarily) mean they mistreat patients. For example, often the inability to leave gets blamed for job, family, and generally the future prospects to evaporate, which in turn causes suicide, which seems to me to find support in the observation that it's most often just after treatment ends that the actual successful suicide happens. The treatment causes suicide, but not directly.

We should make the exceptions that there are plenty of studies about specific institutions where abuse of patients does happen and it is definitely the institution itself that causes suicide.