How to solve problems with the Invisibility power
Obstacles block line of sight and movement and Invisible characters are not considered obstacles, thus people can attempt to move and shoot through spaces containing invisible characters, as they don't "know" the invisible character is there. But that doesn't mean that characters won't affect the results of that action. I believe the "not an obstacle" is there to resolve any situations like "Well you're aiming at Garbonzo, but Invisible Sinistra is in the way so you can't".

Invisibility isn't Intangibility (as you're aware). Characters cannot occupy the same locations normally, and just because one of them is Invisible doesn't change anything. I don't remember if there's anything in the rules that say you can't choose to run into someone (thus forcing a "bump"), but I play like that can happen. So as soon as the Invisible character's square is entered, the "bump" gets resolved as normal, and then the active player's turn continues.

Similarly, as Invisible people are subject to burst attacks and line of effects, I say they suffer the brunt of any targeted effects that were going to pass through the square they occupy to get to the intended target.

But honestly, we just take out Invisibility and Intangibility because we feel it unnecessarily prolongs the game...

EDIT:

At the end of the obstacles section there is a clarification for thrown gadgets which says "Nothing may be thrown through a warrior's square". While this is in the obstacle section (which an invisible character is not) the phrasing of the paragraph leads me to believe it doesn't matter. Rather than saying "Nothing can be thrown through a square with an obstacle and Nothing can be thrown into a square with a permanent obstacle." they specified warrior. However, this doesn't cover ranged attacks from non-gadgets that attempt to pass through an invisible character, such as Zorch Ball. I would say that these other attacks don't "need" to travel in a straight line, so the invisible character gets bypassed.

Ranged attacks are aimed at specific squares rather than warriors. In the ranged strike section, it specifically says that two warriors occupying the same square do not block each other from ranged attacks. So if all involved warriors could be subject to the attack, all of them get attacked.

The rules say that permanent obstacles and living warriors are obstacles. Warriors cannot move into or through a space occupied by an obstacle. Invisibility says that this warrior is not an obstacle. Therefore other warriors can move into and through the invisible warriors space. But the other warriors are still obstacles, so the invisible warrior cannot move into or through their space, at least not without causing a bump. While this doesn't make much sense, that's how it's written.

Then there's the problem of reappearing. Since Intangibility specifically mentions the case where an intangible warrior appears in a space that another warrior occupies and Invisibility doesn't, this means either they forgot about this possibility for Invisibility or they didn't intend for it to happen in the first place. First, occupying the same space as another warrior shouldn't prevent an invisible warrior from appearing; there's no basis for that. 

So there's a couple ways to resolve this


Use the same thing that intangbility does and kill both characters.
Make both characters resolve a bump and make one/both of them resolve random movement
Don't allow this situation to happen in the first place and make bumps occur when any two non-intangible warriors attempt to move into the same space.