Я написала в Edalnice по поводу ситуации, ответ получила следующий:
"thank you for your inquiry; please find the answer below.

In the case of a rented vehicle, the exemption can be submitted by the renter / driver, provided that they have a Power of Attorney from the vehicle owner or registered operator (i.e. the rental company). The authorisation must allow the renter to act on behalf of the owner/operator for the purpose of submitting the exemption registration.

For vehicles owned or operated by a foreign rental company, it is important to note that we cannot verify company ownership through Czech public registers.For this reason, the exemption can be accepted only if the application is completed correctly, including:
• full and accurate vehicle details with proper technical documentation, and
• proper completion of the operator information on the second page of the
exemption form, corresponding to the rental company,

If all required information is provided correctly and the authorisation is valid, the exemption registration can be processed even for rented vehicles
from abroad.

If the exemption is not registered in advance, the vehicle is considered non-exempt in the system, and purchasing a motorway vignette is recommended to avoid potential penalties."

Выходит, что на каждую арендованную тачку (то есть на каждый новый рег. номер машины) надо каждый раз писать в Teilauto и просить их данную доверенность. По сути проблему это никак решает и в правилах образовывается некая лакуна: да, электрички освобождены от виньетки, но освобождение ты фиг подашь, если тачка арендовання.