Infinitive: "to convince a judge on the Foreign Intelligence Surveillance Court"

Question

According to grammar, in the latter sentence, "it had sufficient cause to believe Mr Trump or his associates were involved with terrorists or foreign spies" is a complete sentence. Added with "to convince...Court", this sentence is more hard to understand. So, what is the "to convince...Court" function here?

Thank you!

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If it were shown that his administration illegally snooped on Mr Trump, Mr Obama’s legacy would be disgraced. Alternatively, if it secured a warrant to bug Mr Trump’s phones, that would mean it had sufficient cause to believe Mr Trump or his associates were involved with terrorists or foreign spies to convince a judge on the Foreign Intelligence Surveillance Court.

source: Donald Trump’s habit of making accusations without evidence is corrosive

Answer 1

It's a badly worded sentence.

Alternatively, if it secured a warrant to bug Mr Trump’s phones, that would mean it had sufficient cause to believe Mr Trump or his associates were involved with terrorists or foreign spies in order to convince a judge on the Foreign Intelligence Surveillance Court.

I have inserted the words in order which I think makes it clearer.

Answer 2

The problem comes from the fact that

Foreign Intelligence Surveillance Court

is composed of four words with a formula of adjective-noun adjective-noun where the first three words constitute a full adjective to the fourth word: court.

For this reason for an English learner it would seem a little bit odd, but check this out:

United States Foreign Intelligence Surveillance Court of Review

It seems more meaningful, but it contains the above same sentence. Because of the helping words coming on the prefix and suffix of the original sentence, they give a little bit more of a meaning to our ears.

So it depends on us to accustom ourselves to their listenings.