I can't figure out where you got it from that am R win in GA depends on bigger checks. Who told you that? Some taking head on CNN?
It's hard to believe an immediate $2000pp reward won't stimulate strong Democratic turnout and swing some voters. In the meantime the people voting against handouts were mostly staunch Republicans already.
Have a look at this thread for example. How many die-hard Republicans are clamoring for a Democrat style handout? How many would vote for a candidate that will give it to them, in a State that was decided by 10k votes? Now, you don't need to swing Republicans, a minor bump in D turnout can decide the election. There is no shortage of people who don't care at all about the government but would go out of their way to try to get $2k.