Why on earth bother to ask here and then call UA again? It doesn't matter what you are told, it won't happen.

UA will not transport your bags between any of the three NYC airports. Period. You will find this out when the check-in agent wherever you originate can't print bag tags to your final destination.

The easy solution here is to take your bags with you into Manhattan and leave them with the bellman of any of the large hotels. A very nice tip is in order. It's how they make a living. When you are ready to head out to JFK, head to the hotel, pick up your bags and off you go.

As to getting UA to stick your bags on one of its LGA-VIE flights, you will find yourself waiting a very long time for that !