I don't know if it's harder for women, but I'm sure there's a local team that you can find to tell you about the local scene. A novice (male) racer acquaintance of mine moved abroad this year and wound up being national Macedonian elite road champion. That said, the road season is winding down for this year. So, that means you have a lot of time to figure it out for next year. In the meantime, see if cyclocross is popular in Hungary.

You can get started on a touring bike if you strip it of racks, fenders, etc. It's not going to be the best kind of bike for racing in the long run, though. The geometry doesn't make it the best for fast cornering or accelerations, and it'll be on the heavy side for races with much climbing. I think it will satisfy all the rules, though.