I am currently in my third DBT group in Sydney. I do not have BPD but struggle with distress tolerance and emotional regulation (both modules of DBT). Like you I could not afford to pay for DBT as I have been off work for two years now due to my mental health complications. My first two short DBT programs were delivered by the local university in their gambling clinic. Try calling around different university's that teach psychology, especially Masters in Psychology, there are typically clinics and groups that are helped run by students and their advisors. I am currently in a 32 week program that is solely aimed at clients with BPD (I manage to wiggle my way in), it is run by a team at my local hospital, I did have to wait 10mths but this is the most extensive free DBT program around and worth the wait. Try contacting your local community mental health team, it's great to be linked in with and they can assist with crisis support, they should also have an idea of the programs running in your area.
It's very frustrating having to either pay through the nose or wait for large periods of time to get the services your daughter needs. I'm not sure what is available state by state and what is national, in NSW a GP can write up a mental health care plan that assists with 10 sessions which can be used on psychology, some psychologists may have a gap fee our may not. There are psychologists that do one-on-one DBT and that might be something your daughter may be interested in while you navigate your way though the health system.
Im am currently doing an assignment on BPD for my masters. Having a diagnosis can sometimes bring some relief to your daughter and family. I work with teens and once given a diagnosis most feel like there is a reason for the way they feel and act and feel less alone in what they are going through.
It sounds like you are a great, proactive mum!!!! Whilst your daughter is in hospital it might be a good idea to write down all the questions you have for the team looking after her because it is easy to forget what you wanted to ask them. Also if you are meeting the team (Doctors, psychologists, social workers etc) and you feel comfortable, having someone come with you for support can be very useful . There is always lots to take in and having someone there to listen as well can really help.
I hope your daughter is discharged home safely.
I wish you and your daughter all the best,