I have a little familiarity with firearms and I have never heard anyone call a .22 rifle a twenty-two, always a point two-two, with point being optional. My experience of firearms is limited to the UK and Europe.
If I heard (possibly even if I read) the expression twenty-two rifle I might think it was something like a thirty-thirty rifle, in which the first thirty is the calibre (0.30 inches) and the second thirty is the standard load, in grains, of the ammunition.
Then there is the expression thirty-aught-six or 30-06 which is a type of rifle ammunition. Again, the thirty is the calibre, but the aught-six is the year of its introduction to use by the US Army (1906, rather than 2006).
I think that if OP is writing fiction they would be well advised to get this kind of detail right (and therefore stick to .22 or two-two in this case). To people to whom this kind of verisimilitude matters, it matters. To the rest, they won't care either way.