The 1987 TW's unique alternating current lighting circuit likely requires some intelligent diagnosis from an experienced owner.
Someone here hopefully can help who has knowledge of the '87.
Any more information on the history of this problem? Is the issue new and possibly occurred after some change?
I would suspect poor ignition caused by headlight circuit short. Can a new, proper replacement headlight element be installed as a test?
I would try to inspect all the associated wiring for a possible short , worn insulation, etc.