Request for advice

To my knowledge, oral contracts count just as much as written ones. The problem, of course, is proving that if there are no witnesses. In this respect, the question arises whether you would also admit the agreement in court, then you would probably have to pay for it. On the other hand, the question arises whether the architect should have pointed out to you that it would be significantly more expensive that way. Here one could also quite possibly suspect "nepotism," especially if it is rather unusual for this work to be done by the heating engineer. FYI, with an acquaintance, we as inexperienced floor layers laid a 130m² house with 4 men in one day!!! Ok, there was quite a bit of time pressure, as the screed came the next day and 2 men worked until 4 in the morning. Nevertheless, for example, with us 2 men did that in one day and laid about 170m² (including part of the basement). But in our case fewer intricate cuts were necessary, as there were practically no cables laid on the floor.

Oh please - ignorance does not protect against punishment; in this case: learning the hard way.

The OP was informed by their architect, the OP failed to ask about the exact modalities, did not object, but agreed.

If the case went to court, it would not be cheaper for the OP either, since testimony vs. testimony would stand. Logical consequence: settlement. This means perhaps half the costs for drywall and insulation, but double costs for the lawyer. In the end, those are the costs currently on the table.

Construction expert on the go

Possibly a costly lesson, but that would still annoy me enormously. It should simply be taken for granted that the architect, who works for me, clearly points out that such a change in the order will lead to increased costs. No one can tell me that this shouldn't be self-evident.

That you can clearly see again here that everything, really everything, must be questioned multiple times, I find once again sad.

That doesn't change the fact that the OP will probably have to bite the bullet here. Every alternative is at least just as expensive and will cost additional nerves and time.

I also think you can clearly see the two sides here. Every customer assumes that an architect, whom I have commissioned and who works for ME, really works in my interest and at least points out to me if a service is suddenly going to cost three times as much without any real reason, or if it is even unusual to outsource this service to someone else.
Funny enough, the architect probably "earns" even more from this since their fee is based on the construction sum. A knave who thinks ill of this.
The customer-friendly contractor side, of course, feels no guilt—if "the fool" doesn't ask, we "of course" squeeze him like a lemon.
As I said, 20% more or less would probably hardly upset anyone, but seeing a price increase of more than 200% dismissed as insignificant if the customer doesn’t ask is, in my view, outrageous and would probably also negate any verbal agreements if it went to court.

Very annoying and hard to prove...

What is definitely possible, however, is to demand a performance report with exactly listed hours, who worked on what and when. A craftsman must also be able to do this retrospectively (actually he should do this regularly, but that doesn't always happen) and it would be good, if necessary, to also have such a performance report from other construction sites for comparable work as a comparison. Certainly, an increased amount of time can be justified by special circumstances (more cutting, adjusting, etc. is needed because...), but a 200% additional effort must first be justified.
The justification: he needs longer because he is not that skilled, can certainly only be accepted to a limited extent, because training and familiarization cannot be the responsibility of the client. So if he only needs so much longer because he first has to familiarize himself, then this effort cannot be charged to the client.

I would proceed like this, get a comparable report (maybe someone in this forum has something like that?), then demand the exact performance report from the heating engineer and then you have a basis for discussion.
If he just messes up so badly that a job another person easily does in one hour takes him three hours, then I do think you can negotiate here.

Funny enough, the architect probably "earns" even more himself because his fee is based on the construction sum. A knave who thinks ill of it.

Oh, that's just a typical pub argument – what does the architect earn more if the costs increase by 2000 euros? Always referring to a specific service phase...