After customs there is a drop off desk for bags from international flights that are tagged to another destination. No need to schlepp anything to the domestic terminal. I have done it many times at many different US international gateways and never had a bag disappear or get delayed because of that. Obviously, if you have a really tight connection, the bag needs to get from TB to T6, but given your plan to hang out at a lounge, that doesn’t seem to be a concern.