There are privacy issues that will prevent the doc from sharing most (all?) of the information about your wife's medical history. Yes, both of you should go and settle this in person if possible.
I can understand why they want the amount paid in full . . . payments on a $220 bill are more of a hassle and they just want to get it over with. Of course much of it could be settled if they refiled the claim with proper coding . . .
Your carrier has an appeal process and this is outlined in your policy or SPD (if this is an employer plan). But it appears the issue is not with your carrier, but with the provider.
Take all of your records, including the EOB from the carrier and sit down with their billing clerk. Some providers have off-site services that handle all their claims so before going to see the doc find out who handles their coding and billing.
Before you go you might want to look up the ICD9 codes on the EOB from your carrier. Try this site
http://www.medical-coding.net/sec/icd9/?path=/sec/icd9/ to see if you can get a match on the diagnosis coding. If that doesn't work, Google ICD9 lookup and you will get a list of sites that have free online searching capability. If the ICD9 on the EOB does not match what was diagnosed, there is the root of the problem.
It is possible the paper work given your wife at time of her treatment will have a different (conflicting) ICD9 code when compared to your EOB.
Hope this helps.