Recommendation for calling regular taxi during labor in Japan

Hi,

My wife and I are expecting a baby next year, and the area we are living in do not have a labor taxi service (陣痛タクシー) so it looks like we have to call a regular taxi. Can anyone give me some advise on calling and using a regular taxi, or sharing their experience? I was worried that many drivers will turn down due to concern on hygiene/cleaning for their car.

I'm thinking of purchasing a waterproof sheet for transportation as well, which could help avoid cleaning fee or put the driver mind at ease, is there any recommendation which I should use?

And for infant transportation, should I prepare an infant seat in advance or the taxi (especially around the hospital) generally have that?

Thank you!