I have three industrial machines, one has a lift height of 3/4", the second has a lift height of 7/8", and the long arm machine has a lift height of 1". The Singer 201 is considered a home machine. You can get around your problem by making the fabric stick out past the base of the sun visor. That way you'll only be sewing through a few layers of fabric. If you want to have the visors exactly like original, cut down the bases first.
No one lives forever, the trick is creating something that will.