The "that" here does play a role. It indicates that the following idea is a continuation of Bethany Bell's reporting and understanding of the situation (the 'that' refers back to "Bethany Bell says..,"). If the word 'that' were not there, it would make it possible to understand Russia's insistence as being reported as 'general fact', or as the opinion of the writer of the article, rather than as Bethany's specific contribution. Compare the following two sentences:
Beth said there is no way she could attend, but that John could go in her place.
Beth said there is no way she could attend, but John could go in her place.
In the first sentence, there is no way to interpret the sentence as anything other than being, from beginning to end, about Beth's own opinion.
The second sentence, by contrast, is ambiguous on whether Beth thinks John could go in her place, or whether the speaker of the sentence thinks John could go in her place.