Building permit not granted - Channel overloaded

I would bring the issue to the municipal council. If the city wants increased urban density, the civil engineering department must also create the possibilities for it.

I agree with that. The only question is what this actually brings the OP. You can only hope that the wastewater association is politically overruled or forced to connect. But if the sewer is already backing up now... it probably needs to be renewed, and there's no point even talking about the costs or the timeline. If that's the solution, the OP won't be building a house now, but in 5 years.

Would the plot be large enough for a wastewater tank? Is there access for trucks to empty it? A small sewage treatment plant might also be an option, but of course, everything costs money and it's not exactly "nice" to put up a new building without a wastewater connection...

As Plan C, perhaps also dimension the infiltration for the parental home so that their "rainwater capacity" in the combined sewer is freed up for your wastewater. Of course, in consultation with the civil engineering office.

Maybe two or three neighbors will join in as well. It often saves money if you infiltrate the water yourself. And the areas to be reserved for this are often surprisingly small. I would also say that in such cases direct conversation is often better than correspondence.

Do you have anything in writing from the city regarding the statements that building is permitted in this way? Or were those only verbal assurances?

Was there a preliminary building inquiry before the building application?

What speaks against the construction of a small sewage treatment plant?