Yes, that is technically possible. Mice and humans also come into contact fairly often, so natural transmission from humans to mice (and back) wouldn't be that rare an event.
> and spread it later to human to counter more agressive variant
I think you might be giving too much credit to virologists - I don't think anybody could know from observing a virus in mice if it was going to be more or less aggressive in humans.
> and help that way bring up natural immunity?
Natural immunity doesn't seem to be very useful against covid. Or, at least, not more useful than the vaccines we already have. Stories of people getting covid twice, even three times, are getting more and more common. This doesn't seem to be a "get it once and you're set for life" virus.