Your voltage drops are likely due to corrosion. Have you tried cleaning all the contacts and connections along the way to the headlight?
The way I understand the diagram for the Headlight Relay is that, it cut's the power to my headlights so the starter motor wont have any issue starting the Bike. Or maybe i'm wrong.How long ago was the GPS installed? The mechanic may remember and advise you of the circuit he tied into. I would assume your bike has an accessory circuit that he used. In that case it should be fine. Also the amperage draw of your GPS is likely quite low and might not be an issue even if it was tied into the headlight circuit. In any case, it would be good for you to know which circuit he used in case you have any future wiring problems.
As a general rule it is not safe to bypass a relay. The purpose of the relay is to allow very small wires to be used for control of the circuit, right up to the relay. The relay will then use heavier wires to send current to the actual load which in this case is your headlight. If you are certain that your bypass setup is only on the heavier wires then it should be ok, but in that case you won't have any way to use the handlebar switches to turn the headlights on or off. The final caveat is that without studying the wiring diagram, I cannot say for certain if a bypass can be done safely.