I'll wager an opinion here.
I would bill each of these visits separately. I would bill the sick visit for the pneumonia to the patient's commercial insurance, and then bill the MVA to the auto insurance. It has been my experience that most commerical payers will deny claims for an MVA and will direct the provider to bill the claims to the auto insurance first. Once the auto insurance has maxed on injury benefits, then the commerical payer may reimburse. This has been my experience.
Also, when billing the MVA, I would recommend reporting the appropriate E codes and also get the claim information (i.e. claim number, adjuster's name, etc). You may also have to send in copies of the notes with each claim.
Hope this helps and again this is my opinion.