Took a bit longer than I was hoping, but I finally managed to get the shiny Kyogre from gen 3 I've wanted for a while. Thanks Matt, for offering that helping hand so that I could get my SIDs for my games, which led to me to this
>>33753465, which in turn helped me find my SID for Emerald myself.
It's slightly bittersweet, though, since I initially thought I had a better Modest shiny spread (8/2/20/31/21/31, 10k-ish frame, so 4 minutes), but it turns out I mistyped my TID in RNG reporter. Luckily, this file had ANOTHER solid/pretty okay Modest shiny spread, and an earlier one as well, at about 3 minutes in. Spread was 22/6/24/20/27/27, so I think I might have preferred the other one to be shiny, but this was good enough. I'll probably use this frame/spread for Regice, and then I'll be done with this Emerald file. All of my other shiny spreads are total ass.
Really was a bit of a waste, since I could probably get a better Kyogre spread in HGSS, but mons all the way from gen 3 do have that novelty factor, I suppose.