Deputy leader Keith Brown says it is unfair that opposition parties are invited to comment on Ms Sturgeon's live media conference, while the SNP is not.
The claims were branded "laughable" by opponents, who have previously complained that coverage of the daily briefings shown live by the national broadcaster give the SNP leader an unfair public profile.
The coverage now leaves the question-and-answer session with journalists and switches to contributions from Conservative, Labour, Greens and Liberal Democrat politicians.
A letter from Mr Brown to the BBC focuses on the change of name of the programme to BBC Scotland News Special. It was previously called Coronavirus Update.
Mr Brown states: "The change of title of the programme reflects the focus of the latter part of the programme, which is now, without doubt, political in nature.
"To not feature the SNP in that political space is grossly misrepresentative and deeply unfair."
Mr Brown says the Tory leader Douglas Ross has commented on non-Covid related issues which the First Minister generally does not do at her briefing.
"The First Minister, as any regular viewer will know, goes out of her way to avoid making political points even when invited to by journalists," Mr Brown adds.
"So why is the BBC enabling non-SNP politicians to have a 'free hit’? I urge you to reconsider the format of and our exclusion from this programme as a matter of urgency."
But Tory chief whip Miles Briggs said: “Not content with leading a daily televised briefing, the SNP now want the right to reply to themselves.
“It’s madness and it would be laughable if the result of their pressure and lobbying of the BBC wasn’t so serious.
“They are determined to turn Scotland into a one-party state where only SNP voices are heard. On the current evidence, they’re succeeding.
“We are happy to give Keith Brown the facts. The SNP receive about ten times the coverage of the main opposition party in these briefings each week, just four months before an election."