Only 1 person can own a Etsy shop, as the shop was initially set up with your details, everything has to match your information.
Etsy does not deal well with LLCs. Others have reported dropping the LLC and just being a person and everything is good to go.
@BnCGames: Etsy is not very good at providing properly informative messages so it is always hard to know exactly what they want, but remember that in your case there are essentially at least three separate 'verifications' involved - the account owner, the business, and the banking information. For automated verifications Etsy, or more likely whichever third party vendor(s) they use, want everything to match exactly. Furthermore you may have had Part A verified and Part B verified but since A does not match B they system does not know what to do so and considers it a verification failure.
These forums are full of similar reports of 'edge' cases like yours not going through the process smoothly. You may need to contact Etsy to get someone to look at your case, or hopefully one of the Etsy moderators will see this thread and forward you on to a specialist team.
In any case, please come back and let us know when this is resolved so we can provide the next person who experiences this with better advice.