The problem is, what if they hang up? They have an unfair advantage here since they can refuse to consent to your recording, but you can't do the same thing. (I mean, you can, but then they're holding whatever you need to get done hostage.)
(b) What if they just do all the typical customer service BS on the phone over snail mail? People spend hours or even days with a lot of back-and-forth to get non-routine stuff sorted out... if you do that over snail mail it could easily take many months to do things right. Not exactly practical, especially when you have a deadline coming up and it passes during this back-and-forth mailing.
Is there any legal regulation or contract compelling them to respond to you? If so, use that as leverage. If there is not, you may not be able to compel them to do what you want.