I am in a similar position as you - accepted to Fordham, Hunter, Baruch and waiting to hear back from Teachers College. I too have been weighing the reputation of a school vs practical fit and thought I would share some thoughts I've had.
I met with a friend who recently graduated from Teachers College who informed me that she applied to numerous internship placements and competed for placement among not only her own classmates but mental health students from other schools as well as social work students from her school and others. She landed a "very good placement" but many of her classmates found themselves in less desirable internships. Her take was that her resume demonstrated more experience than many of her classmates who came straight from undergraduate studies and lacked experience in working directly with people. She informed me there were students from less prestigious schools also at her site but like her were older and had quite a bit of experience prior to graduate studies. My take away from this is that the reputation of a school is likely to bolster the resume of someone inexperienced but is not necessary to get a good internship, especially for those with quite a bit of prior experience. Based on your post it sounds like you have plenty of prior experience.
Given that Baruch is such a new program they certainly have fewer connections than other programs do but if you found a site to volunteer at already then it sounds like you're fully capable of reaching out to prospective places and building a connection yourself. I have quite a niche area within mental health where I would like to work so the ability to take such a hands on role in my internship placement is actually a positive attribute of a program but I can see how one wouldn't wish to be burdened with having to take such a hands on approach while also focusing on their studies.
I cannot speak for the graduate program at Baruch but I attended Baruch as an undergraduate in psychology and absolute loved the faculty, location, building, price, etc. It is actually a far more competitive program than Fordham. According to Peterson's the acceptance rate is 36% at Baruch's mental health program vs Fordham' 73%. I am also put off by programs that do not interview potential students given so much of counseling is how you communicate one on one with people!
I have to say it depends largely on what you see yourself doing. If you plan on having a private practice then the name of your school can certainly help. If you're business savvy and know how to market yourself or are willing to pay someone to do so then this matters less. If you plan on continuing your studies after a master's research will obviously be an important part of your resume and Fordham has A LOT more to offer in that department. To me this is the biggest thing that Fordham has to offer over Baruch. However, there are still possibilities at Baruch. But if you ultimately see yourself in a non-profit, clinic, hospital, or school setting I cannot stress enough how NOT worth it is to take out loans for your studies. I was a teacher previously and thankfully graduated without debt but many of my coworkers from far more prestigious schools were swimming and debt and here were working at the same place for the same paltry salary. Personally it doesn't make sense to go into debt for a job that has such a minimal starting salary and low ceiling. I too am considering a PhD after my masters and am also concerned about attending a newer and lesser known program. I have decided to do a concurrent yearlong masters through CUNY Grad Center related to my relatively niche mental health interest in hopes of that rounding out my resume and making me a more competitive phd applicant if I decide to pursue that in the future. Perhaps if you do research while doing your master's this will give you an edge and make up for attending a lesser known program.
I honestly feel that if you're hard working, competent, and experienced you'll be fine regardless of where you go. Good luck!