Is house construction financing possible at all?

I would support that too. Friends live very happily as a family of four in a Flair 113 with almost 30 sqm less than your desired house, completely without a garage. They deliberately chose a smaller house with a relaxed payment rate.

Good tip.
With your stated income/expenses, I would wait exactly 12 months (if the plot remains available).
Then about €20,000 should be saved and your equity would then be at €95k.
Then you can see exactly if you can manage with your money when you put aside €1,700 per month.
Then buy the Flair 113 (or House Erika) for max. €250k. No garage, no outdoor facilities. Nothing at first.
At 6% interest and repayment, that then costs about €1,250 per month.
And then the garage will be built when the money has been saved for it, or the outdoor facilities.
But if the first step doesn’t work out, then you better leave it.

We know craftsmen (also real self-employed) who could possibly do a lot off the books, although I’m still not sure how that’s supposed to work with the bank and presenting invoices, etc.