Here, wild things are mixed that have nothing to do with each other.
Asset management or rental and leasing is to be classified in the area of income taxes. There is no (commercial) entrepreneurial status here.
For VAT purposes, however, the intention to generate revenue is decisive for the entrepreneurial status. The services from the rental are therefore already a taxable supply according to §1(1) No. 1 USTG. However, the rental to private individuals is tax-exempt according to §4 No. 12 USTG and cannot be circumvented by the tax option under §9 USTG. As far as I know, zero declarations also had to be submitted in the VAT return in the past. This is to clarify the matter first.
Now to the specific case.
If an entrepreneur from Austria builds in Germany, then for a service under §3a USTG the VAT liability is at the location of the property. So definitely VAT must be paid in Germany. If the client is a VAT entrepreneur, the VAT liability shifts to the recipient of the service according to §13b USTG. If there is no entrepreneur, the company would have to register in Germany. Meanwhile, there is also the OSS procedure to avoid the registration obligation, but I do not know right now if this applies in this specific case.
Now the lessor is also a VAT entrepreneur. Insofar, §13b UST (shift of tax liability) can indeed apply.
I once had in a seminar that a small business owner must pay the VAT when the tax liability shifts or in the case of intra-community acquisition, but cannot claim input tax deduction (because of being a small business owner). I could well imagine that this is also the case for rental and leasing.
Since no tax advice may be given here, I point out that this is a layperson’s opinion and recommend consulting a tax advisor.