Question: In this case, the sentence reads "You must have made your weekly claim for benefits for the week ending [date] by [date]".

This is referring to the fact that a person claimed a week of benefits too late, and should have claimed them earlier. However, while "must" is generally considered a stronger word, in this case it sounds more like we are saying "We think you claimed benefits this date, but we're not sure." Am I correct in reading it this way? Would the sentence be clearer if it said "You SHOULD have made your weekly claim for benefits for the week ending [date] by [date]"?

Answer: 
I agree with you. Most modals have a **deontic** sense (about necessities and obligations, and effects in the world) and an **epistemic** sense (about people's knowledge about the world).

"Must" is usually deontic when followed by a simple verb,

> 
> You must go tomorrow.
> 
> 
> 

but when followed by the so-called "perfect infinitive" - "have" + past-participle - it is much more likely to be epistemic. 

> 
> You must have been there!
> 
> 
> 

This is much less so for "should": "should have" may be deontic:

> 
> You should have left immediately.
> 
> 
> 

or epistemic:

> 
> They should have got home by now. I'll try ringing them. 
> 
> 
>
This sentence:

> 
> You **must** have made your weekly claim …
> 
> 
> 

is a logical conclusion of what actually happened. We observe some facts, and conclude that you **did** make you weekly claim as described - otherwise the observed facts would have been different. 

If you change it to:

> 
> You **should** have made your weekly claim …
> 
> 
> 

the meaning changes completely. Now it is a statement about what should have happened, but didn't. You **did not** make your weekly claim as described, although you should have.