welcome to the forum!
do I read "Austria" as your home country correctly?
(I am asking as I am also from Austria - Vienna, to be precise - but - while I know that it is a local practice - I had not yet been aware that falconry is a huge part of Austrian cultural heritage, e.g. compared to countries like United Arab Emirates, Azerbaijan, and so on).
Side note - I looked up "Falconry as cultural heritage" and was surprised and impressed to see that Austria was the second country (after United Arab Emirates) mentioned at the UNESCO site as having proposed this as living cultural heritage. So I learned something new today
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So what I mean is only that - in Austria - luckily, it is typically easier possible than in other countries practising falconry to get many products that can help you to transition from a non-vegan lifestyle to a vegan lifestyle.
Examples would include "Beyond Burgers", vegan "Schnitzel", "Wilmersburger" vegan cheese or similar, which are often even available at your local "Billa" or "Hofer" supermarkets, unless you really live in a very remote location. (in that case, mail-order or specialty shops might be an option)
The other good thing is that to make the world a better place, you do not necessarily need to adopt veganism 100% or not at all.
If you are active in caring for predatory birds (which I think is actually a super cool activity!) and it is very important to you, you do not have to give it up immediately, but rather you can at the same time start to eat plant-based food and wear plant-based clothes instead of animal products to make the world a little better.
Take care,
Andy