I've been thinking about this, and here are some Pokemon that can break this core, and definitely Pokemon you'll want to deal with using the other four members of your party.
Breloom resists everything but Shadow Ball and Gyro Ball, and if it's Poison Heal Breloom it can switch in all of the various statuses. Boil Over will usually fail to break its subs unless Buru invests in offense, and then it still won't break the subseeder's subs. Even Shadow Ball usually doesn't 2HKO, while Seed Bomb or Technician Bullet Seed will always at least 2HKO. Buru can run Ice Beam, which can catch Breloom on the switch, but as it's faster and Technician Breloom usually OHKOs, it isn't reliable. As for Nattorei, Power Whip and Gyro Ball will both break the standard subpuncher's subs, although Loom can scout for Whip misses or (in the case of max-HP Loom, such as the standard Gen4 subseeder) run more HP and sometimes soak two Whips on a sub. Poison Heal Loom OHKOs Nattorei with Focus Punch unless it's 252/252 +def nature (and even then it's a coinflip), Techniloom can take a hit on the switch, SD in Natt's face, and OHKO with +2 LO-boosted Mach Punch.
Birijion poses a similar threat. Without Ice Beam on Buru, it's setup bait for both special and physical Biri, and Nattorei will be maimed by either STAB Close Combat or STAB Focus Blast. A fast Taunt puts either Pokemon in a bad position, as well. It has even better bulk than Breloom, and laughs at anything either Pokemon can do but Ice Beam (and even that isn't too hot).
Zuruzukin is bulky, can set up on either Pokemon, and can (and generally will) run Shed Skin to slough off status. It can easily outspeed both, or it can be just slow enough to get double effect from Payback, whichever works better for the individual set. Its offenses aren't anything special, though, so Taunt Burungeru can usually keep it from setting up.
Sazandora can pose problems. It can carry Taunt, can hit both for SE damage, and resists Boil Over, Surf, Shadow Ball, and Power Whip. It's usually choiced and usually doesn't carry Dark STAB, so good prediction or careful scouting of its moveset may help.
Magnezone is an issue for both, unless it isn't carrying HP Fire for some reason. Specs Magnezone is the worst-case scenario because it can muscle through Buru's special bulk, although any set will require careful prediction. Zone doesn't have any way to eat multiple Boils/Surfs, though, if you have some other check to switch in when it comes in on Buru.
Zapdos and Voltlos all have no trouble getting in on Nattorei and can cause a lot of havoc, setting up or getting behind a sub and making life difficult. Neither of them likes Ice Beam, though, and Voltlos won't much like status, either.
Shibirudon, Rotom-H, and Electivire are similar to Zapdos and Voltlos, although they really can't switch into Burungeru multiple times and might not be able to outdamage its recovery, depending on the set. None of them like status, either.
Doryuuzu can set up on either Pokemon and go for a sweep, although it won't like eating a Power Whip/Boil Over/Surf/Ice Beam and thereby losing its Balloon and getting knocked into OHKO range for priority users.
This isn't a comprehensive threat list, just a list of some threats you'd want to consider when making the rest of your team. I thought I'd also mention that, while these both normally wall bulky waters beautifully, Slowbro/Slowking can run Fire Blast to punish Nattorei. It can't do much to Burungeru, even if it does run Thunderbolt, though.