No. He won't need a diagnosis either. It is all down to the restrictions he experiences in terms of PIP activities. These are:
Daily Living
1 meal prep
2 taking nutrition
3 managing medication/ therapy
4 washing and bathing
5 toilet needs
6 dressing/ undressing
7 communication
8 reading
9 engaging with others
10 prompting
Mobility
11 planning/ following journeys
12 mobility
With mental health issues he'd probably be looking at needing prompting, reminding and encouraging with:
Daily Living:
Meal prep
Medication
Taking nutrition - possibly (we're talking weight loss from not eating - snacking doesn't count)
Washing/ bathing
Dressing/ undressing
Engaging
Budgeting - possibly?
Mobility:
Planning and following journeys
He needs to talk about how he is 'most days' (50% or more of the time).
He'll need to satisfy the qualifying period too - so difficulties 3 months before the claim and be reasonably expected to have the difficulties 9 months into the future.
Then focus on weather he can complete a PIP activity: safely, within a reasonable time, to an acceptable standard.
I'd focus on what would happen if he was left home alone for a few weeks without contact from you or anyone else. Would he prepare a meal for himself? Would he wash/ shower? Would he dress/ undress?
In terms of engaging with others face to face - does he avoid friends, social situations, keep his head down if he's out?
In terms of planning/ following a journey. Can he go to places unaccompanied? Can he only go to familiar places alone? What would happen if he got lost or there was an unexpected diversion? The bar is quite high - he'd need to evidence 'overwhelming psychological distress' (basically panic attacks).
What I would suggest is getting is medical records from about the past year to the present date. So a full print out. And sending them with the claim form. It sounds like he's been back and forth a lot. So it'll all be evidenced. He can access them free - just needs to make a subject access to records request.
In terms of ESA. That is more challenging. He may qualify for the support group under some of the mental/ cognitive descriptors. But I would think it'd be more because work would present a substantial risk to him, because it leads to a deterioration in his mental health. It's something called a Regulation 35 argument. CAB may or may not be able to help with that unfortunately. You may be better off seeing out your local council Welfare Rights Service for more specialist knowledge. Medical records evidencing the deterioration would be priceless to help this argument.
Have a look at pip.info and WCA.info. They contain stacks of useful info and will show the scoring systems for both PIP and the Work Capability Assessment.