I just tried to export the zone. I didn't see two overlapping polylines per zone. So maybe this is something, that possible to fix. I set outline for the zone, in DWG translator set to export only zone stamp. As a result - the only one polyline showing zone boundary in AutoCAD.
As about polyline width - this is something very specific (extra feature) in AutoCAD. Maybe it is possible to write AutoLISP code (or try to find it on AutoCAD forums), that will search specific polylines and set the width.
The only problem then - AutoCAD LT does not support LISP. You need full version for that.
Currently I’ve found the opposite process - script is changing width to 0. Maybe by changing numbers in this script can opposite increase the width: