Just so you know you can usually get support for anxiety and depression for free through your university without DSA.
I have dyspraxia, depression and anxiety and have only recently started applying for DSA (and this is just for my dyspraxia, my mental health was/is handled by my universities and the NHS for free). For my entire undergraduate degree I didn't need to use DSA.
Hopefully that is some comfort! Is there a specific reason you need DSA for your depression? (i.e. is there a specific cost you need help with?).
I would also contact the disability services at your uni, it may be possible for them to help you with contacting student finance, provide you with the support you need for free or give you the money to pay for another doctor's form via the hardship fund (though you may need to be matriculated before you can do this).
Is there a partner, friend or relative who can talk to student finance on your behalf? Sometimes it can be easier to be present while someone else actually speaks on your behalf as you can collect your thoughts and feel calmer. They may also be able to argue more strongly in your favour (e.g. the fact that your doctor filled in the form correctly implies student finance should absorb the extra cost and you should be able to make a complaint). My partner handles my phone calls and I find this much easier.