One thing I can say for sure is stay away from anything Sea & Sea. If you already have something from them, sell it. I had a YSD2 and since it was new the battery contacts had issues sometimes and right when I would enter the water it would just shut off, even with brand new batteries. I scraped to all the contacts as clean as I could and it seemed to work a little better after that. Still frustrating for something brand new. Now It is no longer usable after it flooded through the strobe section of it's housing. The battery compartment was completely dry but water was around the bulb. I sent it in to get fixed and after around three months of nothing, they finally tell me they can't fix it and I'm just out $700 dollars. It was only 14 months old and the warranty was only 12 months so they wouldn't replace it either.
I've had the opportunity to use some of Big Blue Lights 15,000 lm and 25,000 lumen video lights. These are very bright and would definitely work for you. The only problem is the edge of where the light stops is very sharp. So if turned on and pointed toward a completely black wall, it would be a very sharp edged circle. I've been looking into diffusers for them that would feather the edge. There is no comparison between these and a 3,000 lumen video light. In fact a 3,00 lumen video light won't do much and is just on the edge of being enough light with such a wide flat spread. All that being said, I would definitely rather have a strobe for still photos than I would a video light. Even compared to the 2,500lm light, I can get brighter shots with a strobe.
I started cave diving a couple years ago and have been working on under water photography in the caves since I started. Not nearly as much experience as some but I can tell you what I've tried and am going to use for myself. Every cave photographer I've seen use Ikelite strobes. I've been borrowing a friends DS125 for a little while and have been very happy with what it's capable of. I've also got a DS161 I recently purchased that should arrive any day.
I've already done the research for you and have been looking into this specific topic for the last two years.I've also been researching for friends to get some strobes to photograph caves within the same parameters you listed. If you can afford it, the DS161 with Li-on batteries is the way to go. Since you're on a budget, I'm going to recommend the same thing for you as I do my own friends. Just get some DS125 and you'll be set. They are by far the cheapest and most durable strobe that will work well for you. I saw a pair of them go recently for around $375 but usually they're $200 a piece. They also have two different diffusers available as well depending on how soft you need the light.
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