Wild mice, hybridization Question.

Okay so, this is embarrassing for me, but I didn't know who to ask.

We have cats.
I have mice.
My mice are completely safe unless they get loose, and also leave my room. 
Back in January, three of my blue tans got out.
The cat got one, and I never saw the other two, so I thought the cat had already...consumed them.
This kind of thing is really rare here, but it does happen once in a blue moon.
Well.

They didn't get consumed.
I've found quite a nasty surprise in my storage room.
One of the blue tans was on top of a litter of agouti babies. 
There is another litter in there somewhere, so I think the other doe is alive too.
So now I have this scenario, where I have two does, two litters and at least one wild male in the storage room.

This is entirely unacceptable on a few levels, so I've been catching them.
I think I've almost got it done. I got the male, and I got mom and one litter. 
Now I just have to get the other one. 

Now, these blue tans are really important to me. 
The last of that line died out in February, or so I thought. 
They are descended from my first blue mouse, bluebell who recently passed away at three years old. 
So I'm wondering, has anyone ever had any success with half wilds?
Looking at these babies, it's weird, but they don't seem to have lost much type. 
They seem to be about two or three weeks old though, so who knows.

In addition to that, would it even be recommended?
The mice I've caught have shiny coats, no signs of distress or illness, no sniffling or wheezing, no external parasites or anything. 

I'm just sort of tossing the idea around.
The male is definatly being relocated somewhere a while away, but the babies?
I could always cull all but my original does and try to breed them to one of my males after they get
quarantined and rested, but if anyone has ever kept bubs like this before it might not be a bad idea. 
My blue tans haven't been exactly prolific, so the litters surprised me. 
I don't know. I just thought some feedback would help a little. 

(I should note that they only get out when there is a undiscovered hole in a cage corner. )

Cull them and then quarinteen the heck out of the doe and breed her to something better. If you don't have any bucks in that line...just wait. I have blue bucks that will be ready next week or maybe a tan buck who will be ready in 4 weeks or so. Don't use the half-wilds. It will take forever to get rid of the unwanted traits and those babies will be nuts.

True.
Eh. I have my does, and that's wonderful as it is. 
The good thing is my mice are all the way on the other side of the house, in a separated "wing". So quarantine should be easy enough. 
-shrug-
In any case, I did get all the wilds out, so no more of that.

I had almost this exact scenario pop up.  In my case, a wild mouse broke into my female colony of extreme piebalds, chewing a hole through the lid.  He was only in there for a day, but several weeks later I suddenly had a whole slew of solid agouti litters pop up.  They were completely indistinguishable from wild-type in both looks and temperament (They bit like alligators), but I kept one of the females as an experiment.  I back-crossed her to one of my piebald males, and she produced a litter of six.  Of the six, three were solid agouti, and three were piebald.  Their temperaments, too, varied.  And here's where it gets interesting.  It is possibly coincidence, but the three piebald mice were distinctly less wild than their agouti siblings.  I tested their behavior enough to prove to myself it wasn't some kind of bias on my part, but a real result.  I have no idea if there is some linkage there or not, but it is a data point for you.  (reminded me of the domestication syndrome observed in foxes) One of the piebald offspring was pretty much just as tame as my usual lot, so in just two generations, it's possible to breed out the wild-type personality.  The hybrids were amazing mouse athletes, btw.  (Which, I found, isn't a good thing!)

I once rescued two litters of agouti and PEW babies whose grandfather was a house mouse.  They had really shiny coats and were strong and fast.  They were adventurous, but tamable.  Several of them lived long lives, including Pladypus' Babaganush and rjduve's Houdini.  Baba sadly passed at two plus years.  Houdini is still going, hand tame and loving his Cheerios and cheek rubs, at three years and four months.

As to losing type, I don't think it would be any worse than having bred to pet store type--maybe even better since your typical house mouse is far more muscular.

I have had this happen twice, don't worry.
 The first time, I kept two does, who I passed on to an interested person. She said they pretty much just escaped and went the way of wild mice.
 This last time, I wasn't sure who the father was, because mom was brindle. Well, I have an agouti female and a very fat brindle female who are definitely "hybrids". They don't like to be caught, but once caught, they are darling and love to brux, especially the very fat yellow one. I am honestly glad I have these two, they make me smile.
 There were three, but I sadly culled one as an adult because she was unhandleable, biting wildly when held.

I think it's a toss up, really. You can sort of tell when they are hoppers, there will be some who are crazy and some who at least allow holding. I was at least told never to keep a male from this circumstance, I guess they are supposed to be insane.