Yes that's right, the relay controls the strobe.
The strobe functions perfectly without the relay, yes - and it turns out that it's not just the strobe that causes the surge. *Any* 120v load on the relay will cause the surge when it turns on.
Yup - the LED lights work perfectly without the relay. And you are correct, when the relay switches on, the LED lights get super bright momentarily while their voltage soars (I have yet to use a meter to get a reading on it, though).
I wonder if a DPDT relay would help (I'm using a SPST relay on the hot wire of the circuit currently).
I'm tremendously curious to solve this issue, but I should tell you that I devised a solution to my needs over the weekend: I'm using two battery-powered strobes with transistors soldered onto the battery leads. I use the Arduino to fully saturate the transistor (at 5v), and tada - it acts just like a relay switch.
Still, if you have any insight, I'd love to hear it.