Is it possible to switch from basic supply to construction power?

You will only need the construction power as long as the final meter is being installed. Whether the construction power meter can simply be transferred to your meter box and you can take the tariff from the construction power box with you, I would first check with the network operator. A fixed-term contract for construction power only makes sense if you can take the contract with you.

With us (EAM, North Hesse), this is strictly prohibited. You will get a new meter.

I wouldn't bother with the stress, not that you end up paying twice in the end

Also here again for the sake of completeness: Yes, it works, you can switch. Self-test successful.