I think the single best piece of advice that I could offer anyone thinking about their first big game hunt in Europe would be to call the Hunting Consortium and have this exact conversation with Bob or Rob Kern. They have been acting as worldwide consultants for a very long time and have tremendous options for you across Europe. If you are interested in turning the hunt into a vacation as well, Margarete can arrange all that as well.
WITH THE HUNTING CONSORTIUM
huntingconsortium.com
I, or any other hunters with experience in Europe, can only offer our relatively limited opinions and experiences. The Kerns have a huge catalog of options that they can tailor to whatever you eventually desire. Moreover, they would be glad to simply discuss possibilities in the very general way you have outlined in your post.
I would note, you can easily hunt a red stag in Eastern Europe in the price range you have listed.