If you gave the bank accurate and truthful information and they've seen your income and outgoings then it probaby is sustainable... I'm guessing you're in London?
It really depends on what the rest of your lifestyle is like.
I probably 'need' no more than 800 pounds after essential bills and I have a very, very comfortable life that some would say is luxurious (nice hobbies, eating out whenever I fancy, memberships, upmarket clothes, nice car). However I'm a single woman, no kids, no obligations, live in a cheap rural area and don't have to worry about emergencies or paying for anyone else.
You need to look at what you guys spend money on and also have an emergency fund set aside. You may well be ok on 2k a month depending on your lifestyle.
Perhaps your wife could do with a job though, even if it is to have a more comfortable lifestyle, overpay the mortgage, be able to support your child once they are 18 etc.
Imagine her having a basic 2K pcm role and being able to put all that money away... Takes the pressure off you, too. And can be seasonal or temporary.
EDIT: Just seen your income and outgoings.... well if that's the case it doesn't sound sustainable! 🤔Are you able to re-budget? That's a large deficit there!