Can I use a video light that turns off automatically when a photo is taken with a strobe?

This is an auto-off feature that exists mostly on focus lights, which are used merely to focus on a subject.
When the strobe fires, they turn off for 1 second to avoid a hotspot on the photo.

However, focus lights are usually very weak with a narrow beam and aren’t designed to shoot video.

Luckily, there are a few different possibilities.

One option is from Sealife; they are now offering proper video lights with an auto flash off feature.

This would be an excellent choice:

Or the more affordable Sealife 3000:

One example of this type is available for the Kraken lights:

The remote is situated close to fingertips so you can turn off even 2 lights quickly with a quick press and without moving your hand, then turn them back on as needed.

Ikelite makes several models for this: including here and here.