I feel for you, OP. I've been where you are and it's hell.
Self-certify for now and get an appointment to see the GP. I'd be very surprised ifb they didn't sign you off.
I'm also shocked at your manager's attitude. They have a duty of care and should be supporting you, not washing their hands of any responsibility. It might be ducks in a row time and I would start a paper trail.
If your workplace is unionised, speak to your union rep. However, I rather suspect that it isn't.
If the company is big enough to have a stress management policy and someone in HR who deals with H&S, check out the policy. Then email your manager pointing out any bits that are relevant in terms of what action your manager should be taking. Also email asking for an appt with whoever in HR deals with this kind of stuff.
If there's no such person/policy, have a think about what your manager could do to reduce the pressure on you, and suggest it, again by email. If he/she replies face to face, follow it up with another email summarising the discussion. Avoid being bolshie and using legalistic language, but it make clear that work is making you ill.
Also look up the judgment in the case Walker v Northumberland County Council (a landmark legal ruling in stress at work). It won't be terribly relevant but it makes it clear how serious workplace stress is and employers without a dedicated HR team often have no idea about its implications.
But definitely take some time to recoup. I struggled on for a month when under intolerable pressure, and ended being signed off for 4 months. It was the worst time of my life. Taking time out now could well nip it in the bud.
And when you do, make sure you take time to do nice things - go for walks, swim, see friends/family, whatever will help you reflect and recover.
And definitely start a timeline of what has led up to this, when you have raised issues with management and what their response was. This could all be relevant later on.