Would the weather be a problem driving between Auckland and Christchurch in late September?

mild/cold

I assume you realize that you can't drive between Auckland and Christchurch, you must fly or take the ferry.

That said, you can expect just about any kind of weather in NZ in September. Generally speaking, it will be colder the further south you go. Where specifically are you planning to visit?

Allow for rain too. I've had that in the middle of an NZ summer. Drive down to Wellington, drop your car off, foot passenger across and the same car company will have another car for you in Picton (assuming you tell them and give a date).

It may rain and be cool or cold, but nothing weather-wise should impede the trip.