I see two main issues for you to try to address:
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the lack of support for you with you constant caregiving (which would be exhausting even without all the extra difficulties you are facing)
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the constant crying and likelihood your baby is in pain
You have already had a lot of good advice about what might be wrong with the baby but it sounds like you are facing a fight to get the GP/HV team to take this seriously.
This will be so much harder when you are sleep deprived and under so much stress.
So I wanted to see if I could give a few ideas to help address the first problem.
Firstly, your partner. You say he is around on Sunday? That is only two days away. As an initial emergency measure, can you ask that he gives you a break on Sunday morning where he takes the baby and lets you sleep/have a hot shower/bath? If you know you have this 'booked in' it gives you something to look forward too.
Family - is there anyone who will take the baby or come round to give you moral support for any length of time during the day or night? Even just to make you a cup of tea or hold the baby while you drink it? Failing that, would anyone take your other children for the weekend just to allow you and your partner to focus on yourselves and the baby?
Do you have any neighbours who would help? Come round and hold the baby while you shower/have a cup of tea? (We had older neighbours who offered to help and I know I could count on them)
Do you have any local friends who might offer support? Any local Facebook groups you can join? I have found my local positive birth and breastfeeding groups supportive on a number of much wider issues even though you are past the birth/bf stage.
I think there are some brilliant CMPA/I facebook groups where I am sure you will find lots of people going through the same thing - worth joining these.
Do you have a local children's centre? Head there tomorrow if you do. It is likely someone there will offer help and support or talk you through local options. Do they run any courses/groups/training sessions you could attend? Free baby massage or sensory class which you could try and see if they can help the baby relax? The techniques might help, but equally you might find that the staff or other parents you meet are a bigger source of support.
What about turning up at the next local baby weigh in clinic? If your HV isn't helpful, see whether another member of the team is more helpful. In the early days when my baby was constantly crying I turned up and a nursery nurse saw us. She instantaneously offered to book an appt to come round and teach me some baby massage techniques for free. This really helped, just knowing someone cared.
Can you check out if you have a local Homestart charity? Contact them to see if anyone can help you? Also check out PANDAS Foundation and see if they have a local group in your area.
What about through your student midwife course? Any local fellow students who are perhaps young/single/childless who might be willing to offer you some support to supplement their training?
Given you say both the GP and HV have been so unhelpful, I wonder if they feel you are coping due to your job and experience, or do you put on a brave front when you meet with them? From your posts it seems like this is really starting to affect your mental health so perhaps see if they can offer you personally any extra support, although be very clear that your symptoms are only as a result of living with a constantly crying/nonsleeping baby for months without support.
Whenever you speak to anyone about your situation. Explain upfront about your professional experience as a student midwife and as a mum of two, and emphasise that all this experience tells you that what your baby is going through is not normal. Sometimes HCPs will (wrongfully) dismiss parent concerns as being over exaggerated so you need to use everything you have to make them take you seriously.