Hi Grace,
As you have already highlighted implementing a new screening tool or referral algorithm is likely to be challenging as it will be seen as additional provider time per patient in an already stretched environment.
Have you thought of exploring an automated telephone screening system for initial screening for depression? This may not work for suicidal ideation but could perhaps work for screening for depression. This could be managed by health coaches or nurses and depending on how patients respond to automated questions, they can be triaged for a call back by a nurse or doctor. There maybe more buy-in from providers if they don't have to spend time doing the initial screening.
I am not sure how feasible this would be but maybe something to consider. If interested in exploring, I would be happy to share information on the automated telephone self management system which is usually used for self-management of chronic diseases.
-Purba