I'm a bit lost by this taxi driver placard thing. I think somehow a private car company/limo service is getting confused with a taxi.
In my experience, taxi drivers only get out of the car to put things in the trunk. I don't think one would be standing around with a placard.
I suspect that, in the absence of someone here actually having experience ordering a cab in advance, it's not done. I'd expect that they would refer you to a car service. I'm certainly no expert, but I haven't found most Japanese companies to be especially flexible. Generally this is what they do, and no matter how similar this other thing is, it's not what they do.