Can you do any diy? With that budget you could build a very potent milti sub setup that would best most any retail sub except those the level of the Pb16 U ,JTR 118ht or Seaton Submersive and maybe some of them.
Ported or sealed comes down to your goals. Do you want the lowest extention into the deep ULF bass with a tight sound that requires lots of drivers/power (sealed) or do you want maxim spl with less drivers and power required but extension limited to the port tune of the enclosure(usually between 15hz and 20hz sometimes a little deeper with corner loading). Sealed takes more drivers and power to reach the same spl as ported because of the natural roll off of the sealed alignment. Eq can help but that's where the more power comes in as boosting with eq eats amplifier power.
Your room is large including the connecting rooms(like mine) and I would recommend going ported and no less than dual ported 15"'s or 18"'s as it would require close to 4 sealed subs and double the power for the same output(but again with lower extension and less group delay which helps with integration and to some yields a tighter sound)
Diysoundgroup has both ported and sealed flat packs for 15's and 18's that only need to be glued together and then finished in your choice of finish be that Duratex/paint or fancy veneer. Parts Express has sealed flatpacks too as well as the drivers for the ported and sealed boxes at Diysoundgroup.
2 solid Diy options within your budget or just over would be
(Prices are close estimates)
(Ported) 2 Diysundgroup ported 18" cubes ($520 w/shipping), 2 Dayton audio Rss460 drivers($520 free shipping) and a Behringer Inuke 6000Dsp ($400 free shipping from Parts express ) finish and needed accessories ($80) Total=$1520. Wired one 4ohm driver per channel
(Sealed) 4 Dayton audio 18" Ultimax flatpack bundles from Parts Express ($400 free shipping) Behringer Inuke 6000Dsp ($400 Free shipping Parts Express) finish and needed accessories ($125) Total=$2125 Wired each drivers dual voice coils in series then 2 drivers in series for a final load of 8ohms per channel ( about 700 watts RMS per driver) then as funds allow you could add another Inuke 6000dsp and almost double the power (about 1050 watts per driver) by running one driver per channel at 4ohms.
Inuke will need fan mod to be quiet if it will be in the room with you but its easy and there are many threads showing how here on AVS,
Now if you cant Diy 2 of the (Ported) JTR 118ht's ( one now and a second later) or a (sealed) master and slave pair from Seaton sound would be my recommend. Either choice would let you try one and add another as needed or as funds become available. Both are close to your budget and regarded as some of the very best outside of the Diy options.
Its not that I don't like SVS or think they make great subs, they do. Its just that you have a good budget and can look at what I consider better BOOM for your dollar options like Diy or JTR/Seaton sound for that kind of money. I don't recall anyone who has gone with either of the Diy/JTR/Seaton offerings that weren't elated with the performance.
A great sub system can last through several speaker/receiver upgrades so go for broke now on the subs and enjoy the fact you did it right for years to come, possibly for life.
No matter what path you choose the main thing I would stress after countless upgrades to my own sub system is to push your budget now and do it right the first time as I have spent way more upgrading than I would have if I had just bought more than I thought I need from the start. Big rooms take BIG bass and nobody I know of ever regretted going overkill when it comes to the subs. What you don't want is to be running at 85%-100% of your subs performance all the time to get the impact you want. Headroom is your friend.
Choose a path. Sealed or ported/ Diy or retail.
Go overkill/ More than you think you need.
Enjoy ultimate performance from a sub system that never breaks a sweat even when pushed to Reference plus volume and cruises along for many years to come with no desire to upgrade.
I'm no expert. Just my thought's on the subject. Good luck. Your welcome to PM me for more DIY info if you want. I have built several Diy subs now and feel confident I can help you get the impact you are looking for if you can Diy.