when you say 'custom' export task - did you create this yourself using python, or within arcmap maybe?
It sounds like you are close to a resolution in that the output directory stamping is incorrect.
You do get an image generated, it is stamped to disk, its just that the response has an incorrect url.
I work mostly in js not silverlight so this may not be as applicable to you.
you have two approaches as i see it - you try to fix the output response or you workaround and rearrange that url when it is returned.
For the fix, maybe you can recreate the custom print tool the way you did it originally but with a design for its external use. Part of the issues with gp tasks, publishing and ags is that esri has a single solution for desktop and server. But sometimes you are creating tools that have no desktop counterpart or are never intended for the desktop. yet you must go through arcmap.
For the workaround - if the url is in a predictable format, and you have a chance at accessing it in code before it is resolved then maybe you could use regex or something similar to modify that url and correct it.
With the js api there is a print task widget that makes it easy to call print jobs. But it also takes control away from the developer. In this instance you want the control back so you can recreate the print calls yourself and not rely on a widget so that you can access the response url and reorder it the way you want.