A government minister has the same right to ask the court to hand down a particular sentence as you or I do. We can. Ask but will be completely ignored. The Home Office, nor anyone else, has any authority to "recommend" any sentence.
Life sentencing is set by statutory guidelines and the court has to follow them.
I don't see why Ministers would be being kept "well briefed" at all. Its just a criminal trial and, as such, has nothing to do with government.
The idea that the government will be asking the court to impose any "restrictions" is completely daft. They won't and any minister seen trying to influence the work of a court will have some very serious questions to answer
The suggestion that anyone involved with this case will be receiving communications as to what course of action is "desireable" to government is ridiculous, conspiracy theory nonsense, quite frankly!