I almost always belong to the "budget fits" faction because we built relatively cheaply ourselves and got quite a lot for our money (but we also live very rurally and prices here are noticeably lower compared to the city!). But even I would say here that it is very unlikely to be possible to build a finished house including everything and with little own contribution in a moderately expensive region within the budget. We ourselves built our house including garage for under 350,000 euros including the land, all incidental costs, and the necessary exterior work (sowed lawn, laid terrace), with a good portion of own work. But our plot only cost 35,000 euros including all incidental costs. And the kitchen with 13,000 euros plus furniture, etc. was also on top! In addition, we built with an architect and individual contracting because this is regionally cheaper here than a prefab house builder/general contractor; but in the cities, it looks different... And you really shouldn’t underestimate that building a house in the countryside is significantly cheaper.
I consider a budget of around 220,000 euros (excluding land) for house, all incidental costs, exterior work, and also kitchen and such to be unfeasible. Of course, it depends on your own expectations, but I’m afraid even if you stick to the absolute minimum equipment (meaning 120 sqm living space, no garage/carport, no underfloor heating but radiators, cheapest heating system, no electric shutters, no walk-in shower, cheapest sanitary fittings, little electrics, no ventilation, etc.), it could be tight. And then there is the question of whether you really want that for yourself in your own new house...
If in doubt, the only help is: check out companies and providers and get 1-2 quotes, like from the usual suspects Town & Country, Heinz von Heiden, etc... Also look at regional smaller providers; whether it’s prefab or solid house is initially secondary (the rumor that prefab houses are cheaper is persistent). How skilled are you yourselves at handiwork? Would it be possible to do some of it yourselves? And I don’t mean laying floors and painting walls. Would it be possible to have a shell house constructed and finish the interior step by step yourselves or commission those trades you can’t do yourself from companies? That saves money but involves a lot of (organizational) effort, time, and nerves, and can really go wrong in the worst case; it’s just a question of whether you want to take that risk...
What is your income like? Is increasing the budget by about 50,000 euros totally out of the question?