I would buy a knock off 1747-CP3 cable or build your own. See if you can configure RSLinx to talk to it through the serial port using DF1 protocol. I suspect you have a better than 80% chance that the serial port is set up for DF1 Full Duplex and you just need a simple serial cable:
https://www.plccable.com/allen-bradley-slc-1747-cp3-controllogix-1756-cp3-10ft/
If you do not have a good quality USB to Serial converter, I highly recommend this one:
https://www.plccable.com/usb-to-rs232-plc-programming-cable-special-plctalk-net/
You should not need to buy the expensive DH+ cable and should not need the DH-485 (UIC) cable unless you are sure the 5/04 has its serial port set up for DH485. If you really believe you need a DH-485 cable then buy this one:
https://www.plccable.com/allen-bradley-sale-1747-uic-usb-to-dh485-usb-version-1747-pic-slc-500/
Get online and save the program. Go offline and change the controller type to the specific model of SLC 5/05 you need. Then Save As.. with a new name. Do NOT clear the I/O or resize data tables when prompted.
Set the channel configuration for Channel 1 (Ethernet) to match what you will want. Save again. Install the new 5/05, with the key switch in program mode. Power it up. Using the same serial cable, establish communication to the 5/05 using RSLinx. Download the modified program into it and switch to RUN mode.
Now you can connect to it with Ethernet.
Changing controller types is pretty easy with various flavors of SLCs. If you do not have a commented copy of the file for the 5/04, you will not have any address comments, but you can build them up from electrical drawings and (if any exist) HMI/SCADA tags that access it.
The serial port could be set up for: DF1 Full Duplex, DH485, ASCII, Modbus, DF1 half duplex, DF1 Radio,...and I think I am forgetting one other possibility... If it is set up for ASCII or Modbus, RSLogix cannot connect to that port. Borrow somebody with a DH+ cable for an hour to make you a backup file to dump in your new SLC... If it is set up for DF1 Radio protocol or Half Duplex Master or Slave, setting up RSLinx to talk to it can be tricky but is still possible in most cases.
Really, if you have a reputable integrator you can call in for an hour, it should take less than than amount of time to connect and produce a RSLogix500 file suitable for your migration to a 5/05.
I have bought all the cables in my links above and they are all superior quality, delivered fast, for much MUCH less money than those with the A/B logo on them.