The 1761-NET-ENI's serial port is hard-set for DF1 Full Duplex point-to-point, and is only meant to be connected directly to a controller's serial port that is also set up for Full Duplex point-to-point.
The Net-ENI considers itself Node 0, and considers the PLC to be Node 1.
So you can't connect a Net-ENI to a 1770-KF2 or 1771-KE or 1785-KE and "browse" or connect to a multipoint network like DH+, because the only possible target address is DH+ node 1.
You could still use it in a system like you describe, though, where the PLC-5 and the 1771-KE are the only nodes on the network. The PLC-5 would be Node 1 and the 1771-KE could be any node number other than 0 or 1 (it's functionally transparent).
The "Classic" PLC-5 models like the PLC-5/25 don't have a Channel 0 serial port. That 15-pin port is actually a DH+ port, and is wired internally to the 3-pin Combincon connector below it. The top two connectors are DH+ only, and the bottom one is RIO only. They can't be configured for either protocol like the Enhanced/Ethernet/ControlNet PLC-5 controllers can.
You will have to carefully research and build cables to go from a 1761-NET-ENI Mini-DIN8 serial port to the 15-pin RS-232 port on the 1771-KE, and another from the 15-pin Data Highway port on the 1771-KE to the 15-pin (or 3-pin) DH+ port on the PLC-5/25.
A 1756 chassis with a ENBT and DHRIO would still be my go-to devices, especially because it could later host a 1756 CPU.
The ENI + KE would add some latency to every packet, but it should still work.