A convicted felon cannot own/possess a firearm in any state under federal law. It is a federal felony crime. Each state has it's own state laws governing restrictions on firearms/handguns/rifles, etc. Check with a criminal defense attorney in your state.
I would not think a compound bow/crossbow would be prohibited because they are not "firearms" BUT I don't practice in TN. Check with a criminal defense lawyer in TN to be sure. Good luck.
I am trying to give you a general answer to your question. We do not have an attorney-client relationship by this response on the avvo website. I have not been retained to represent you. I am licensed to practice law in Kentucky and in federal court in this state and the Southern District of Indiana. You need to seek legal advice from an attorney licensed to practice in your area..