Theoretically, Biden could do it and not be prosecuted.
But if he ordered a member of the military to do it, they are required to refuse illegal orders. I don't know the rules about illegal orders but I bet this would fall under that. At the same time, the President can pardon people convicted in military court so that's not much of a deterrent.
Similarly if he ordered a civilian (say, CIA) to assassinate Trump, that person could be tried. But again, the President's pardon power makes federal charges not much of a threat.
BUT — the President cannot grant pardons for convictions in state courts. So anyone involved would be in trouble if it happened in a US state. And if the Supreme Court did not make the President immune from state-level prosecution, Biden could be tried for being involved... but it seems unlikely that they would go for "the President is immune from federal prosecution but not state prosecution."
Of course, all this show how insane and dangerous the idea of Presidential immunity is. It's a terrible idea.
In theory that gets around the legal issue but then you're getting into practical issues: Trump is protected by the Secret Service so either it will be extremely difficult from a practical perspective, or you would have to get quite a few people to go along with the conspiracy. Again, this highlights what a terrible idea immunity is because the possibilities get horrifying really fast.
I am super duper NOT an expert but I believe the President can pardon crimes in DC, so that would get around the state court issue. Same for outside the US: I'm not sure states can punish crimes outside of the US. Even if they can, they may not have laws on the books to handle that sort of thing.
According to the arguments currently being made to protect Trump from prosecution. The premise of the question was "If presidential immunity is absolute."
They argue many things, but very few of them really hold up to real life, so I personally don't put too much weight on them any more. I trust that the legal system will shoot those arguments down if not fast, then at least efficiently. IMO it is basically just is a method of extending these cases for last as long as possible, and I am surprised that nobody can do anything against that!
NSA is civilian, they work closely with CSS which is the military side. The determining factor for civilian vs military is whether the people working there are enlisted soldiers/commissioned officers or civilians who just get hired like other jobs.