Where can one complain about an architect?

I would only pay net!

Because in this case the offer violates the Price Indication Regulation.

In principle, final prices including VAT must be indicated to end consumers in accordance with § 1 para. 1 Price Indication Regulation.

Accordingly, the subjective obligation to indicate prices applies to anyone who commercially or regularly offers goods or services to end consumers or advertises them by specifying prices, unless the exceptions of § 9 para. 1 nos. 1 to 5 apply.

If the architect complains, you can discreetly point out the following: Incorrect price indications can be sanctioned as a violation of the Economic Criminal Act (f § 10 para. 1 no. 1 Price Indication Regulation) with a fine of up to €25,000. That would be more expensive than waiving €1,400...

Whether gross or net does not matter at first, it should be directly apparent from the written offer or then the invoice. That should be undisputed, right?

One can only try to reclaim the money through a lawyer, but the question is on what legal basis? Incompetence cannot be held directly accountable and the guy did work somehow.

If you talk about net with a private person and do not communicate it clearly, then he is even less worth anything.

but that's the way it is. We had the same issue with the general contractor and markups. After the first incident, we always asked whether it was gross or net. within their invoicing, they always speak of net