AIBU to take sick leave when I am completely exhausted?

Take the time off. You really need to rest and try to recover.

If you take a week off, what would be different upon your return to work that would mean you can cope though? Your studying, children and husband working away would still exist. As would the vomiting unless you currently have a diagnosis and treatment underway.
I’m not saying don’t take the week as you clearly need to but you also need to spend time thinking and putting in place something to mitigate this as you’ll be exhausted all over again and in the long term that’s neither good for you or for your job and employment prospects.

Since you actually sound to be unwell, I wouldn't feel guilty about this. AFAIK you can self-certify for the first week of any illness, so do it. You're sick, so don't feel guilty and ignore your 'D'H's ignorant comments. No job is ever worth making yourself ill (or even iller) for.

You have medical issues and are feeling exhausted as a result. Totally reasonable to take time off. In fact if you can't keep food down and are anaemic presumably you aren't going to be able to perform properly at work anyway. Frame it to work that your absence is due to your inability to keep food down and anaemia and associated extreme fatigue, not just being tired. I would imagine that your GP will happily sign you off once your ability to self certify expires.

Are the investigations proceeding swiftly?

It sounds like you are actually ill though? Not just tired with normal work stuff but otherwise completely healthy.

Unless you're in probation or your company is going through redundancies or something similar that might mean going off ill is counted against you, I'd absolutely do it. It might even be better for your work long term if you came back after a week and were able to give 100% rather than struggling along not being able to work properly for months.

He doesn't sound brilliant to me. Sit at a desk. What a prat.
Only you know how this would affect your employment, but I think going sick might be a great idea.