Pelly, you have an advantage re: meat quality in Europe. Why not get your meat from a butcher? It's better quality than supermarket meat, and not as expensive as strictly pastured. Friends in Europe tell me their butchers supply excellent quality meat that is not as pricey.
Does Hungary still require that organic meat be 60% grass fed? If organic is better priced than strictly pastured, that's a good option too.
Not sure where you are located but I saw this:
You can try the Tőzs-Ért kft. at the Feny utcai piac, they are selling szürkemarha. Based on what I found that's the closest to grass fed organic you can find. Another great butcher is at the Fővám tér market, Gál József.
If these are not options for you, here's what our family did, as we learned about the lack of nutrition and - in many cases - the contamination of most supermarket food and meat.
We took our meat budget and applied it to pastured/grass fed meat only, with the goal to obtain the best nutrition for the money.
For example, instead of supermarket chicken 3 nights per week, we have a farm raised whole chicken (sometimes a lamb or beef roast) on Sunday nights. Leftovers for many days afterward, loosely following this idea
1 Chicken, 17 Healthy Meals...Obviously, the more people in your household, the more your mileage will vary.
Once all the chicken leftovers are used, always save the carcasses/bones to make bone broth, then soup. Or freeze multiple carcasses and make one larger batch.
How to Make Bone BrothBeef marrow bones also make excellent broth, easily made in a crock pot. Have alone for breakfast or lunch, or saute lots of onions and throw them in the pot, top a bowl of it with your favorite bread, broil with cheese and you have the perfect French onion soup. This is a favorite in our house with a big salad and glass of red wine. Beef or chicken bones can be purchased at your local butcher or pastured meat store. If you follow the directions in the link, you can make more than two batches with one batch of bones.
On other nights of the week, we'll have poached pastured eggs for dinner, placed on top of toast (homemade bread) that is lathered in humus (and hot sauce!), or we'll have various lentil recipes with organic or pastured sausage or hot dogs, or perhaps sweet potatoes with kale and bacon, or pastured ground beef in a stir fry with lots of vegetables, and of course, fish but we buy fresh caught, whatever is on sale that week.
(be careful about which fish you buy
LINK)
When you find a good sale on fish or meat, buy a lot and freeze.
And be creative with eggs - there are innumerable recipes for delicious meals.
I'm told there are a ton of recipes out there on how to eat nutritiously on a budget. I'm just giving you a few examples.
I wish I could help more, but I'm not the cook. I'm sure others on the forum can provide better tips.