The jumper wire makes the lights work - would it damage anything to ride around like that?
Bill,
Not really. The idea of the relay is allow the power to the headlight to cut out when you hit the starter. So you don't drain the battery as much with a combination load of headlight and starter.
So jumping it will give you lights but is harder on the battery during starting.
You said you tried the old relay that came with the bike and the lights worked, why not put that in?
More testing ideas..
With ignition on, relay out, put a test light on female plug 85 which is black and from there to battery ground. Should get illumination. yes/no? If no then check black wire which goes next to the starter relay.
Female plug 86 should be brown and should be a ground. One way to test would be, relay out, ignition OFF, run test light from battery positive to 86. Should illuminate. Basically you are forcing power backwards through circuit, but it will only illuminate if the ground wires are good. If it doesn't then there is a broken ground somewhere. This ground joins up with all the other grounds from headlight, dash lights etc.
You do need a relay with the 87a, because 87 and 87a do opposite things. Think 87 gets power when 85,86 is closed and 87a gets power when 85,86 are open.
If Bobs relay has the right plug layout, all except that it is 87 instead of 87a, you should be able to go to a NAPA , show them the relay and ask them to get you an identical one but with 87a.
Sorry I am so long winded.
Good luck.
Chris