Adding the Net-ENI to the mix takes a step back; its EtherNet/IP <-> DF1 feature is only capable of a point-to-point DF1 connection (like when RSLinx is set for SLC-CH0/Micro).
The only workaround would be to configure the PanelView for DH+ Node 1, so that packets from the Net-ENI would be directed to the PanelView only.
But then you wouldn't be able to communicate with the SLC-5/04, thereby making it a moot point.
BUT.... the newer (Series D, I think) 1761-NET-ENI are capable of the more conventional "COM Port Redirector" feature. That will let you use the DF1 driver in RSLinx, restoring your ability to change the "Device Type" to a multi-address DF1 browser.
Other less complex RS-232/Ethernet interfaces (Digi, LanTronix, Moxa, etc) also perform that COM Port Redirector function.