Your scriptural interpretation is bogus. “Render to Caesar that which is Caesars”, means you follow the laws of your country and take part in civil govt.
The bible doesn't belong to Caesar.
In Canada there are several ways of AFFIRMING you will tell the truth .
1. You may be asked to swear on a Bible but if you decline, they will ask if you affirm to tell the truth.
2. Muslims are asked to swear on the Koran
By swearing on a book, you are DECLARING that
you believe and hold precious what is in that book,
So why would a Catholic swear on a protestant bible? (Our courts actually have both).
You wouldn't expect a Muslim to swear on a bible, so no Catholic should swear on any book they don't believe in.