What I have found here on page 2 is that you are allowed to hunt with gun permit + up-to-date "hunting ticket", or with foreign hunting permit. On page 1 there is reference to a law saying that you may strike with your hunting gun or hunting bow if you clearly identified the game and the strike would not endanger the life of someone. The two may cover the case you are interested in.
Here are the laws summarized. It says you need a holding permit for a hunting bow.
Here it says that you can hunt with a bow in Hungary, but for big game (deer, wild boar etc.) the bow must be above 222,7 Newton pulling strength + appropriate arrows (whatever that means, I am not aware of the terms, sorry). Also says that crossbows are not allowed.
Here mentions that you need an extra permit for hunting with bow or with hunting birds, above the average hunting permit.
Disclaimer: I am not a hunter, I just speak Hungarian. I think the best would be to directly contact some hunting clubs or officials. Here are the addresses of the national + district level hunting administration organs. This is a list of hunting clubs. If you write them, try several addresses and don't expect a quick answer :) Hungarians (and generally the older people) are not so good English speakers, so it might take a time for them to react or to find someone to help you.