If that's the case, I would think even more that the Republic would have stopped them in the Core Worlds if they would even try to raid them. You can't destroy city worlds this way (the Mandalorians actually occupied Taris and the Tarisians formed a resistance until the planet was freed by Republic forces).
I think the Republic would resist for quite some time, but by this time the Mandalorians would have heavily outnumbered the Republic (by means of conscription) and would have many more supplies available to them. Once they hit the Core, the Republic would only delay the inevitable. I have no doubt that they would put up a good fight, but they would lose eventually.