Hi Mark,
Sorry I was not clear in my first post (was a bit late when I posted).
I meant going into the Slabs menu, Edit, Hide Edge, and clicking just once where the 2 overlapping edges exist. This should hide just one of the edges of one of the entities, leaving the other which I believe should then look correct in the live self xref since there is only one edge displaying. I could not get a pair of slabs to display the actual problem you were having but I was able to create co-existing edges and perform the hide and verify that the self xref will display only those edges that are set in the drawing space to "SHOW".
EDIT: I was looking at your rendered iso view a bit more close and realized that the entities you might be talking about might not actually exist in the same plane (is one higher above the other?). So my original suggestion might not work for you if you need the aligned edges to show for your rendering. In that case, I do not have a suggestion for you in this scenario, sorry.
Since linetypes are currently applied based on the start of the entity (I believe that is the case?) and not a fixed global or individual point for specific entities, I don't think your wish for a selectable origin point would be realistically possible. I would guess that because of how linetype definitions are created, I am not sure if being able to define a different start point for an entity's linetype definition would work, because similar to how end correction works for linetypes, I think you would end up with a "solid" looking line for the part of the "entity line" that would come before your custom definition start point. I like the idea of being able to set a specific start point (which if a possible solution was found, it could possibly help with the solid line end correction issue), but I think that would be a HUGE task to get working and not sure if it could be translated to DWG for example which would be important.
I For overlapping 2D lines, if they are the same length, linetype and spacing, I have a little button I use to rotate lines 180° By Center, so that the start point for the lines are both at the same end witch aligns the linetypes. Works well also to quickly flip upsidedown text based linetypes like - - - GAS - - - linetypes.