Well, anyone who's seen my other posts knows that for $500, I'm a big fan of the Rythmik FV12. IMO, it's the best $500 sub out there and a true taste of "good" bass all around. Below that price point, there are subwoofers that certainly aren't bad, but they clearly compromise in one or more areas. The FV12 is the least expensive sub that I consider to be good all around with no obvious weaknesses.
That said, I'd be a little bit concerned about its output capabilities in your setup. If your theater room were closed, I would say it's no problem. But with an opening to another room, you have to consider the entire space. Then again, for $500, you won't find a lot of subwoofers that can play a whole lot louder without distortion than the FV12 anyway. One very nice thing about the FV12 is that it is very well engineered, with limiters and filters that make it all but impossible to damage it. So you can push it as loud as you want. It might not be able to deliver the loudness that you're asking for, but at least it won't damage itself and it won't go into crazy distortion. Its design prevents bad things from happening
The HSU VTF-2 MK4 and the Outlaw LFM-1 Plus are like sister models. They are good alternative choices to the FV12. I personally like the FV12 just a little bit better, but it's really close. And any of those three is an enormous step up over your Sony!
I honestly know nothing about the Cadence subs that have been mentioned. I'll have to look into those! Can't offer any input on them though. Sorry
If you go with a Rythmik FV12 or an HSU VTF-2 MK4/Outlaw LFM-1 Plus, you'll get a genuinely good sub and a taste of "real" bass. As I say, with that opening to the kitchen, I'd be a little concerned about maximum output. But if you think your Sony is ok, you'll be blown away by any of those three

You might want to add a second sub or upgrade to an even more capable sub one day. But for right now, my opinion and recommendation is that $500 for the Rythmik FV12 is about the best you can do.
Best of luck with your upgrade!