Wait with the building application because of the child construction allowance

Yes, you need to! It only works one after the other. It also says so on the page of Bayernlabo

I have overlooked it until now. Well, of course, we could take our time with the notification of the new place of residence.

You just have to post here and in the afternoon the notification arrives

see

So I looked it up again on the Bayern Labo website.
For the Bayrisches Baukindergeld Plus it says "The application can be submitted from the time you move into your first own home up to a maximum of three months after the date of the payment confirmation from the KfW at the BayernLabo." So there is no 6-month deadline to be met.
For the Eigenheimzulage the notice is not required, only for the regular monthly child benefit.

Just as information. The processing of the home ownership allowance and the Baukindergeld Plus in Bavaria currently takes 8 weeks.

Unfortunately, the application is not as comfortable/digital to submit as with the KfW – meaning fill out online, print, and send with PostIdent.
Furthermore, 4 documents differ: The registration certificate is an extended registration confirmation, the child benefit notices must be sent along, KfW confirmation for Baukindergeld, and proof that you have lived in Bavaria for more than 1 year is required!

After 4 days, you receive an email that the documents have been received.

On Friday, these figures were published in Die Welt

"Indeed, more than three quarters of all applications relate to the purchase of existing properties, i.e. condominiums. Only a quarter of the applications for Baukindergeld actually concern new construction"

as well as

"Most parents applying for the subsidy are between 25 and 40 years old, and 60 percent of them have a taxable household income of no more than 40,000 euros."

Those who are actually building new would therefore belong to the 40% who have an income of more than 40,000 euros.