I find that way too tight. I also think 2,300 EUR is not enough for a family of four to live on comfortably. Sure, it's enough in normal months. But what are normal months anyway? There are almost always some special expenses (insurance, new car tires, house expenses, the children's class trip). From the 2,300 EUR, you also have to save. If not for a vacation (which can definitely be considered dispensable), then for retirement provision, a new (used) car every 10 years, or something as small as a broken washing machine. Also, although the house isn't old, there could very well be some costs coming up in the next 20 to 25 years. What are those supposed to be paid from?
The interest rate won't be exactly great with such minimal equity either.
How high are your current heating-inclusive and heating-exclusive rents?
If the heating-exclusive rent is roughly in the range of the loan installment with 3% repayment, then you might perhaps dare it. If the heating-exclusive rent is significantly below that and you’re not even in your twenties, you should ask yourselves why you don't have more equity (I also hope that you’re not completely drained with the 35k, but that there is at least a roughly five-figure amount available as security). Because then, despite the (significantly) lower "cold burden," it wouldn’t have been possible for you to save substantially.
I can understand that this may be your only chance to get into homeownership. But it will be many long tight years.