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I have encountered this sentence:
Universities should have to record sexual violence claims, says expert.
Like the OP of this question noted, I believe we could choose either should or have to.
So what's the need of using the phrase, instead of the above two options?
EDIT: After some googling, I've found that should have to could mean should be expected to or must have to. Are there any other possible meanings?
I think in some contexts it makes perfect sense, for example if evaluating a requirement put upon by a third party. But it is a nuanced way of speaking.
If I say "you should have to clean your room before being taken out for ice cream", I'm saying that there should be a requirement that you do the cleaning to get the reward, where perhaps that requirement is not in effect. but maybe I'll put it into effect and am just thinking out loud. Maybe my kids will take the hint and clean their rooms without it needing to be a direct order. Perhaps I'm saying that in earshot of my spouse to see if they agree that it should be a requirement. Or maybe I'm opening it up for discussion and my kids will offer to complete another chore instead and I will find that agreeable.
If I tell my kids "you should clean your rooms before going out for ice cream", I am stating it as a request that might be interpreted as a firm requirement, or maybe if you don't I'll be annoyed and only buy you one scoop. Or clearly I want the rooms clean, but if you promise to do so immediately after the ice cream I'm OK with that too. So there is some ambiguity on whether it is a firm directive, but clearly it is something I want to happen.
If I tell my kids "you have to clean your room before being taken out for ice cream", it is a simple directive. No clean room, no ice cream.
Should has more than one meaning here: in the first case it is from WILL meaning 'intend to,' 'bring my whole will and intention to bear.'
You should clean your shoes. // You have to clean your shoes.
When it is combined with have to in this sense it means "there ought to be a general rule that..."
Everybody should have to read Boethius, Magna Carta, the Gettysburg Address.
In your example, which is similar, should should be taken as an ethical assertion, "it is right that they record;" and have to as the legal requirement "they must be compelled to record." This is a tautology only if all laws are morally uncompromised, and all moral obligations are enshrined in law. (Link to Financial Times)
There is also a possibility that 'should' may used to suggest a remote possibility, a hypothetical situation, 'should' is modal.
If I should catch you taking sweets, I should have to tell your mum.
Have to in this context means be required to.
Should in this context is using Merriam-Webster's second definition:
2—used in auxiliary function to express obligation, propriety, or expediency
Your example sentence could therefore be written thus:
It would be proper for Universities to be required to record sexual violence claims, says expert.
This particular construction pretty much always has this meaning. It can also be used to draw attention to something that is not being done, but the writer believes ought to be done:
Police should have to report all violent interactions with the public, but they are not.
Actually, it might be it. I’ve recently come across this thread on some twitter account about ‘Stress’ and look what I’ve found. So, in this case I’d consider this particular example as ‘there ought to be a general rule that...’ 
For me, should + root verb = Suggestion, urge, sharing opinion ... similarly, have to + root verb = Suggestion, urge, sharing opinion... so if should have to + root verb means same as the meaning of 'should + root verb' or 'have to + root verb' then, using 'should have to' is not necessary.
We shouldn't use the phrase "Should have to" since we can use either "should" or "have to". eg. I should go or I have to go. However if you used the phrase "Sholud have", then you have to use "past participle"after this phrase. Eg. I should have gone. but not I should have to go
