Invisibility is supposed to dispel the moment you interact with something (activate a switch, open a door, etc.). Couple that with Oblivon's psychic guards that are capable of divining your exact location no matter how invisible you are when you get a bounty placed on your head, and you really do seem to be in a pickle. If he sees you approach, it really doesn't matter if you are invisible or not, the moment you get tagged with a bounty, you get a massive *YOU ARE HERE* arrow attached to your head for guards to see.
All I can recommend is either try a high-magnitude Chameleon spell (it won't dispel the same way as Invisibility) or improve your Sneak skill.