I would think this should be possible. The mechanism would likely be similar to the suspected mechanism for how SIV originated, which is that it came from two different retroviruses that infected a single host.
The Radiolab episode titled "Patient Zero", starting at around the 35 minute mark, discusses how this is expected to have happened (transcript abridged for clarity):
Chimp zero was hunting and he comes upon a monkey called a Red Cap Mangabey. Eats it. And in the process he catches a red cap mangabey version of the AIDS virus.
Next - some time after that first kill - weeks, months, we don’t know - maybe it was the same day - chimp zero comes across another monkey. And this monkey was called a spot nose guenon. Chimp zero eats that monkey and gets a spot nosed guenon of the AIDS virus or the SIV virus inside it.
So [now] you’ve got the red cap mangabey and you’ve got the the spot nose guenon. Two completely different kinds of SIV viruses inside the same chimp.
Now - under normal circumstances - both of these SIV viruses would go nowhere. They basically are entering a completely alien habitat. The cells don’t look the same. The environment is different. The chimp’s immune system would normally kill them.
But then once in a blue moon, something crazy happens. These 2 viruses will end up inside the same cell in the same chimp at the same time. Literally there is a single cell - simultaneously infected with both viruses.
So suppose on one side of the cell you’ve got the mangabey virus. And on the other side of the same cell you’ve got the spot nose guenon virus.
You have an enzyme. It’s called the polymerase enzyme that’s copying genetic information of the viruses. This is what viruses do. They hijack these enzymes to make copies of themselves.
Now - here’s the problem. These enzymes. They’re not necessarily that sticky. While they’re in the process of copying one virus, every once in a while they’ll accidentally fall off mid copy, and then latch on to the second virus. And just keep on copying. What it ends up spitting out is a hybrid.
Now this new mosaic probably won’t go anywhere because 99.9999999 percent of the time - when these hybrids happen - It’s a dead end. The chimp’s immune system is pretty sophisticated. It has evolved defenses against these viruses - and it will destroy them. But once in a blue moon...
So this is a blue moon after a blue moon after a blue moon to really get this. Finally you get one particular mosaic virus. Between the mangabey and the guenon. That through sheer random luck - works. It landed on the exact right combination of genes that allowed it to evade the chimp’s immune system.
This seems like a good model for how your crossover might happen: A single individual, infected with similar-but-different versions of a virus that its immune system is capable of defending against, but a copying error merges the code of both viruses and---unluckily for the host---ends up making a viable novel version of the virus that the host cannot defend against.
Mice aren't really known for eating other vertebrates, so consumption as a vector seems unlikely. But coming up with such a scenario doesn't seem too implausible.