My problem with this question is that it presumes that Federal employees or elected officials must be "sworn in." The rabbi who was m'gaier me, Rabbi Bertram Leff, said that I should avoid taking "oaths" or "swearing" whether in becoming a Federal employee or being a witness in court. Fortunately, Federal law gives me that option. The text of the "oath of office" is:
"I do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge the duties of the office on which I am about to enter: So help me God."
Since I can "affirm" there is no shavua here.
The second assumption that bothers me is that one is required to swear on a Bible or some other religious tome. Most Federal employees are sworn in without a Bible, because there is usually a room full of us taking the oath at the same time. The use of Bibles is for photo-opportunity occassions, like the swearing in of an elected official or a Presidential appointee. Also members of the military often use a Bible for the photo-op of their taking the oath on commission, enlistment or reenlistment. I've attended some such swearing in ceremonies -- the last one for an Orthodox Jew who was an assistant secretary of Treasury. As I recall, he used a Tanach -- but he just as well didn't use anything. Since there is no obligation to use a Bible there is no issue about choice of religious text from the Federal point of view.
Similarly, Federal and state courts permit a witness to declare or affirm, rather than swear, that they are testifying truthfully or risking penalty of purgery or making a false statement to the Government, whichever is applicable. Given that, use of the court's Bible, or substitution of one's own is irrelevant.