Here is how you were probably wired with the 501. One cable from the satellite dish (you never said if it was a roof or a ground tripod).
One cable from the 3-4 Output to an input of that switch box. That switch box fed the signal to the front TV using coax and to the bedroom TV using coax.
With the 625, you to have the 3-4 cable from the 501 moved to the 3-4 cable of the 625.
You will need to add a coax from the CH 21-69 to a new inpuit of that Distribution Switch, probably AUX or VCR IN.
Then if you push the button on the Distribution Switch for TV2 and whatever Input you choose, the 625 TV2 signal should be going to the bedroom TV. Since the receiver output is label 21-69, I would check the modulator to see it is set for a channel in that range. Set the bedroom TV to the same channel. That should get a display on the TV.
Now some realities. You only have one input cable. Depending on what dish you have, that will limit you to only one tuner working A roof dish you will be limited. A ground tripod might be able to feed both tuners. That is a later thing.
If your front TV has more inputs than coax, component cable, I would recommend that you connect the front TV directly top the 625.
Once you have a display on the bedroom TV then you can work on the remote. The remote is not necessary for the bedroom TV to show anything. That can all be controlled by the buttons on the TV. You should be able to get the bedroom TV to the channel you selected/checked in the modulator setup.
With the remote in sat mode, then you should be able to change the receiver channel, but then again, the second tuner doesn't have an input so that is basically a mute point. But you should be able to view something recorded. Until you have a second satellite input, you probably will run the 625 in single mode.
Depending on the length of your RV, the UHF have have some issues getting to the receiver. RVs can have more interference than a house. That is why I asked about taking the TV2 remote to the receiver and see if it functions. Usually just holding the TV2 remnote is the doorway of the bedroom should work. If the distance is a problem, there is a fix but you need to get signal to the TV first.