Grammatically, why does it seem that 'that' can't follow the verb expressing propositional attitude in this sentence?

Solution 1:

The relevant clauses are ‘the petitioner must submit his own account of the events’ and ‘he claims (that) the events justify the exemption’. When they are joined, they become ‘the petitioner must submit his own account of the events that he claims justify the exemption’. ‘The events’ is not repeated, so no second ‘that’ is required. If you consider that to be a relative pronoun, then it is the object of ‘claims’ and represents ‘the events’. Some linguists consider that not to be a relative pronoun in constructions like this, but a subordinator. According to that analysis, that still introduces the relative clause, but the relativized element is omitted.

Solution 2:

I have not seen this referenced anywhere as such, but I can think of no way to construct counter-examples, either.

The answer is very simple, actually—it does not even have anything to do with repetition or relative pronouns at all:

A complement clause without a subject must not be overtly marked.

To explain more fully:

A complement clause is a subtype of noun clauses, themselves a type of subordinate clauses. Complement clauses begin with the subordinate marker ‘that’ and function as the object of a variety of verbs, mostly verbs that indicate feeling or reporting verbs.

The subordinate marker is nearly always optional in English:

I think that my husband is having an affair.
I think my husband is having an affair.

This is a lot like in relative clauses, where the relative marker (pronoun) that/who(m)/which is optional as long as it is not the subject of the clause:

The man who is walking on the street. [who = subject; not optional]
The man [who(m)] I see walking on the street. [who(m) = object; optional]

The rule of when the marker is optional is a bit different in the complement clauses, though: there the rule is that if the clause has no overt subject (meaning that the actual subject of the clause is located outside the clause, in the main clause that the complement clause is embedded in), then the marker is not optional, but actually disallowed. Compare the following sentences where I also show that the presence or absence of another that (the relative marker) is not relevant. The relative markers (interrogatives in the last section) are in bold, while the subordinate markers are in italics.

With relative marker

The new car that I hope (that) my cheating husband will buy me.
My cheating husband that/who(m) I hope †that will buy me a new car.

Without relative marker

The new car I hope (that) my cheating husband will buy me.
My cheating husband I hope †that will buy me a new car.

In other types of sentences

What car do you think (that) my cheating husband will buy me?
Who do you think †that will buy me that car if my cheating husband won’t?

In the first sentences here, the complement clause is (basically) my husband will buy, which has a subject and a verb. In the second sentences, the clause is will buy, which has no subject.

Since there is no subject, overtly marking the complement clause with the subordinate marker that is impossible.

Note that the clause does of course have a logical subject, but it is not located in the clause itself. It is, in fact, the relative marker that/who(m). This leads to problems if using who(m), because the poor pronoun ends up being the object in the relative clause (whom I hope), but the subject in the embedded complement clause (who will buy)—and it becomes pretty much impossible to decide whether to use who or whom. This is a case where English grammar simply collapses and there is no ‘right’ or ‘wrong’.