What could go wrong if I use my claw hammer as a cross peen hammer?
Well, first off, because the hammer is utterly the wrong shape, it won't work very well - and you can substitute "at all" for "very well" without being very off the mark, especially since you are evidently new to this and don't understand it much at all.

Peening a scythe properly is a process that takes some tools and some skill. With certain tools, more skill is required - with others, the tool design helps to reduce the level of skill needed. For a novice user with limited hammer skills, a special peening tool which uses a pair of caps that run on a central stake over a small anvil to do the actual peening is a common choice (yes, they do cost money, such are tools.) That reduces the skills you need to have to hitting the cap with a hammer (hard enough but not too hard) while holding the blade flat to the anvil and moving it along as you hammer.

Hammering freehand you need the cross peen hammer to concentrate the force of the blow into a small area so it will actually move metal (that's what "peening" is.) But now you not only have to hold the blade flat on the anvil (do you have one?) but you also need to place those hammer blows accurately at the correct distance from the edge (which the fixture above does for you by design.) A claw hammer's broadly rounded, mostly flat face will not hit the precise area needed (only) and it will move very little metal as the force of the blow will be spread out over the wide face.

And you will need a scythe stone and a holder to keep it wet for getting the blade truly sharp after peening, and keeping it that way as you work. I stone about every 5 minutes (takes 20-30 seconds) while working, and peen daily if working a lot, or when I feel like the scythe has done a days worth of work over several days, or when it's had a few too many run-ins with rocks or the like and/or just seems to need it.