The question is arriving at the need for the line down the center of the hexes. To even have the decision of a hexside do the hexes occupied by the attacker and target have to be exactly intersecting by the same line?
The LOS should be from center to center, and only one possible line. Every hex the line enters, no matter how small, counts. if the line only crosses the line between hexes, and doesn't actually enter either of the two hexes it crosses, then only one of the two hexes counts.
If a line goes through a hex and appears to be touching the line between it and another hex, the another hex doesn't count. It has to go in to the hex. the rule about going down the line only applies when the LOS doesn't enter either of the two hexes on either side of the line.
Technically, the LOS can only appear to be touching the line anyway, or its touching the entire line (and therefore would not have entered either hex). Mathematically it can't do both. If it's clearly in one hex, it's not on the line between the two hexes. Lines don't work that way (connecting to a point beyond them).
(I'm not sure which is worse, thinking my explanation is a mess, or worried I missed something about this issue I'm not getting).