I'm afraid we do have customs to enter France and Germany, they are done on the UK exit points, such as the ferry or train. However, they are simpler than travel by air where specific mention of an airgun needs to be made for baggage handling purposes. One could say that unless they checked a vehicle, that they would never know... however the legislation for France and Germany requires an European Firearms Pass (which we can't get).
Interesting that you have EU police and Hungarian Police, we only have the one here.
Perhaps we can raise this on the WFTF email, with those countries, and they might be able to advise what will cover us if we moved through their countries, however to date, i've been told it's either permission explicit to activities in that country from that country's police/firearms control, or a EFP.
It might be worth getting an example of the paperwork ready so those countries can check and get authorisation.
We have a letter from the Home Office, it's a simple photocopied letter, but I'd rather not rely on that at a random border crossing check on a rainy night
Our letter simply states we do not need an EFP, but I would hate to argue that with a foreign police force.
That said, I suspect most will fly direct to Budapest from the UK, as the drive is quite considerable, and flying is cheaper than the petrol you would use in a car getting to Hungary.