Just an idea but how about a simplified, easier and cheaper to produce, as well as more reliable Tiger that becomes as robust and reliable as the Panzer IV?
The Germans are able to develop a vehicle that they are able to produce in sufficient numbers to replace the PzKpfw IV in the heavy companies of each panzer battalion, with the Panther (hopefully similarly simplified) replacing the PzKpfw III in the medium companies. We can sort out the where's how's and why's of the back story later, as well as other details such as assault guns and tank destroyers etc. but the basic thought is an earlier iteration of the Tiger makes the Germans painfully aware of the issues that needed to be fixed in 41 and they actually fix them through 42 and 43 getting them into mass production by mid 43. Anyway the allies encounter the early vehicle in 42 / 43, make the same assumptions they did in reality, then when they land at Normandy run head long into a couple of thousand improved Tigers that have effectively replaced the PzKpfw IV in many Panzer Divisions.
What would this Tiger look like? Pretty much the same except for the road wheels, instead of interleaved dish type there would be a more conventional dual medium or large wheel type seen on many other nations tanks. Most the other changes would be out of sight but their effect would be even more important as they facilitated the improved performance and ease of production.