I have an interest in science but I'm not all that clever. I'll give this a try though.
Firtstly. Faster than light travel is not currently possible as far as we know.
Secondly. I think that "general relativity" comes into play with the headlights.
So for a stationary observer you would probably be invisible. Or maybe just a flash of light.
However since you are travelling at the same speed as the ship....... the ship and its headlights would appear to be functioning normally to you.
For example if you were in the back of a pick up truck doing 200 kph and you fired a pistol out the front of the truck. The bullet would appear to be travelling at normal speed relative to your point of view. However, somebody standing on the side of the road watching you would think you were crazy....... LOL, sorry could not resist that...... They would think that the bullet was travelling 200kph faster than normal.
Hi, you are employing wrong sciences here.
Firstly, nothing can travel faster than light, it's simply not allowed by this universe (maybe there are universes that do allow it but not this one)
A photon (that does travel at the speed of light) when born is immediately transported to it's destination , that is the speed of light.
Let me elaborate, when a photon is born in our Sun, everything and everyone in the universe will agree that the photon is taking time, or traveling untill it reachers earth and get's absorbed for example on your face, warming you on a sunny day.
Same thing with photons that are born in the other side of the galaxy which are currently taking thousands of years to reach us, THE OBERSERVERS, judging by our point of reference.
But, for the photon, it's life span is completely different. It is born on the other side of the galaxy, and is absorbed by it's destination (for example your face)
immediately.
Therefore, not only it is not possible to travel faster than light, a statement "faster than light" is a logical fallacy, it's almost the same as saying "faster than the speed of 3 Isaac Neutons divided by 7 Conan O'Briens", it's as gibberish sentance, as you cannot build a craft from particles that don't even exist for longer than a split moment.
Theoreticly it's even impossible to build a craft that will travel at the speed of light. (however on paper i think it still possible to reach something like 99,99%, don't quote me on this as I do not remember now)
However, there is a way to cheat this, by employing Alcubierre's Drive theory, which is not technicly fictional as it has been proven on paper (the same way that Black Holes were proven years ago but were considered fictional untill their discovery by LIGO in late 2015)
Alcubierre drive theory states that you can build a craft with an engine that uses energy to contract spacetime itself ahead of the craft while expanding it behind the craft :
Allowing the craft to ride the wave of spacetime (as if a surfer would employ a surfboard to ride the ocean wave) and achieve superluminal speeds.
However, here's the interesting bit, the spacetime within the warp bubble would remain unchanged, therefore technicly the craft itself would not be traveling faster than light, even though the bubble itself would be, so yes you can have your headlights shining without the need of breaking the laws of physics.