I think the Bible taken as it is written shows Saul wanting to kill David is out of jealousy and knowing that he can take the kingdom:
1 Samuel 18:8-9 And Saul was very wroth, and the saying displeased him; and he said, They have ascribed unto David ten thousands, and to me they have ascribed but thousands: and what can he have more but the kingdom? And Saul eyed David from that day and forward. (followed by him first trying to kill David with a spear in a few verses down).
God gave (or probably more "allowed") the evil spirit upon Saul (1 Samuel 19:9 And the evil spirit from the LORD was upon Saul...) so the evil spirit was part of God's plan for achieving His purpose.
I do believe in allegorical interpretation in studying the Bible. Understanding that verses are true history but also provide a deeper spiritual message when compared with other references in the Bible (The Bible is it's own dictionary).
I do think there is a picture of Satan vs. Christ in this, or perhaps false spiritual leaders not giving the Lord the kingship (as we see throughout the old testament in prophets being rejected, Christ being killed, and church prior to the reformation, and now in our world polluted end times church. But I have not studied the book Samuel allegorically so just speculating.
I don't think there is anything in the message of this story that specifically supports that it is Satan trying to stop Jesus from being born though.