It took months before I stumbled across the answer to this, so I thought I'd post it in case it's useful to anybody else.

I'm a comparatively new owner of a 2003 LS430, and every time I attach the handicapped placard to the rear view mirror I say bad words, because it is a real struggle. I have never had this problem with another car, with the others it just slips easily over the rear view mirror connector.

I discovered by accident that holding the placard exactly horizontal lets it slip on and off easily, Mine is a RI placard, but maybe they are all the same shape for other states.

 

My significant other needs to use one and when we're driving my car (01 LS430 lux) I slide the placard up into the gap at the top of the windshield and the headliner about 1/2". It fits perfectly and allows it to hang just like if it were from the rear view.

Not to mention it doesn't rub on the rear view and cause any other issues with the sensors or electronics.

Hope this solution works better. I know the mirror stalk leaves little room to be used.

~Andy

 

View from the inside, stuffed between headliner and windshield.

View from the outside.