House search - Construction and purchase of single-family house / granny flat

Good idea. Go to the bank and say you have no equity, want to finance alone, and need >300k€. You realize yourself that sounds absurd, right?

The banker also wants to know that child support will soon have to be paid. I would at least not keep that a secret.

Girlfriend gets nothing from me because she herself is employed full-time.

I don't know how old your child is, but if they are under 3, her income is considered supplementary income. This is usually only taken into account at half, and she would still be entitled to childcare maintenance from you. And in addition to the payment amounts according to the Düsseldorf Table, you can also proportionally calculate additional needs such as kindergarten fees. You should not ignore these amounts in your calculation.

Maybe this doesn't really belong here - but since your private problems were mentioned:

You don't want to move to another apartment in the meantime ... Maybe you're deluding yourself about the timeframe ...

Finding a property - with many compromises 3 - 6 months, with demands easily twice as long

Getting a property - seller's market !!
Even if you want it - others want it too ... - or a compromise dump with few interested parties

Being able to move into the property - still occupied / renovations or refurbishment ??

I would reckon with a timeframe of 6 - 12 months if you really want it and make compromises.
With demands rather 1 - 2 years

Whether the girlfriend will keep her feet still that long?!?

So I would first clear the slate privately. Making such high-priced decisions under time pressure is never good.

Just my 2 Cents ...