Good evening @EBrooks,
Thanks for reaching out and welcome to the Rogers Community!
I understand your desire to operate your DVD player with the Champion Remote. We're glad to know you were able to successfully pair most of your electronic devices.
However, I'm not sure the eject function is something that would be officially supported by the URC2125 Rogers universal remote.
Please check out the Champion Remote user's guide manual.
The Rogers remotes often have limited functionality with other equipment. It also depends on the make/model of the other equipment as to the functionality provided - some do nothing more than turn on/off. You are lucky that you are able to use the remote to control the player. As for the Eject, since you need to be at the player to insert or remove the DVD/BD, it shouldn't be much of a problem to use the eject button on the player itself. If it doesn't have one, keep the player's remote near the player, but most of the players I've seen have an eject button.
One other thought with the Champion remote set to DVD, try pressing various buttons to see if any of them are mapped to the eject functionality. You may be lucky and one of the less-used buttons may work.
If you wish to have full functionality of all your equipment with one remote, then I suggest getting a Harmony remote like the model 650, which often goes on sale for around $50 (Black Friday, Boxing Week, other sales, etc) This is an activity-based remote which turns on equipment, switches to the correct input, etc all with one button press like "Watch TV" or Watch DVD....
I'm well aware of, but usually hesitate to give out information here, regarding the JP1 interface, because it's well beyond the average user, and may make the remote unusable for Rogers functionality, if used incorrectly.
So, user beware, as mentioned by @Pauly
I would like to thank everyone for their responses. I found it quite interesting.
To the person that mentioned that ejecting implies a visit to the actual device, it may be worth noting that the reason an eject button would be helpful is because the controls for the player are on the top of the device and not on the front. The device is sitting on a closed shelf, and it isn't readily apparent where the controls are.
@EBrooks : Thanks for the feedback. Perhaps you could cut a small piece of a yellow Post-It or something similar and put it on the front of the player with an "up-arrow" indicating where the eject button is on top. You might also put something on the eject button itself for a tactile feel, similar to Braille (a blob of nail polish or similar?). This probably wouldn't be necessary for anyone used to using the device, however, it might be useful for others.