The Genie DVR is considered a "hub" of a whole home solution and has 5 internal tuners that receive the signal from the sat's, just like your current HD receivers do. The mini's that they most likely want to "upgrade" you to do NOT have their own internal tuners (why they are so small) and will NOT work on their own without the genie on the same setup because the just use one of the 5 tuners in the Genie to function. They also have their own issues and can be much "touchier" to get them to work correctly, especially in an RV setup.
So if you still want to upgrade to a Genie and be able to have all your receivers function independently of one another, no matter their physical location, you could just replace one of your current HD receivers with a Genie and keep the other two you already have that contain their own tuner. That way, no matter what combination of receivers (genie or HD receiver) you take with you, the others would continue to work at home. Also know that swapping out ANY of your current receivers would put you back into a 2 year agreement with DirecTV, if that matters at all.
Disclaimer: Although you "can" do this, just know that you are technically breaking your agreement with DirecTV if you use your service in the RV AND have someone at home watching as well. Even if it is on a temporary basis, it is what DirecTV considers account stacking and it is against your user agreement which could result in among other things, cancellation of service. [\end paranoia disclaimer].
We have a Genie HR44, 2 HD DVR's (HR24 and HR21) and three standard HD receivers (H25's). I usually take our HR44 Genie and one H25 for the bedroom/outside TV and our other DVR's at home continue to record and operate normally as do our other H25's even though no one is there to watch them.