I haven't noticed any time difference between warranty and non-warranty repairs. The dealer gets paid in either case - either by you or the manufacturer. Sometimes there is a slight delay if a warranty repair has to be approved by the manufacturer, but that doesn't usually take very long.
On the other hand, if you are going to have the problem fixed anyway, regardless of who pays, why not go ahead? You can always argue over a warranty reimbursement later. Aux-man does not want to lose the use of his toy and clearly that time is more important to him than the money. Hard to argue with that.
However, if an Extended Warranty is involved that is a whole different story. Failure to get pre-approval is grounds for denial of coverage under most extended warranty policies.
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL