I am confused slightly by your working. You said the "cover over the reverse bulb only comes out about a quarter of an inch". At least on my early w115, the reverse bulb lens is integrated into the rest of the taillight. So you would need to remove the entire lens. Your choice of words made me wonder if your clear reverse lens has disconnected accidentally from the rest of your lens.

My other thought was if you had some sort of bumper overrider in the way. I do recall they can make changing the taillight bulbs harder. I think you have to remove the overrider in this case.

Anyway, for my early w115 (with no bumper overriders), you just unscrew the plastic round nuts inside the trunk and then the lens can be pulled off. It can get stuck with a bunch of filth over the years, but if you got it to move a quarter inch, then the whole thing would come apart at that point.

Good luck.

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1968 220D, w115, /8, OM615, Automatic transmission.
My 1987 300TD wagon was sold and my 2003 W210 E320 wagon was totaled (sheds tear).