The Banner of World Dragon is vs magical attacks, not non-magical, so perhaps that is a typo. Here are my usual tactics vs the HE bunkers.
Phoenix Guard - Scar vet cowboy with 1+ re-rollable and OTS that is within range of the slaan can pin the front of the unit without taking much damage even if they have the AP banner, allowing a ranked unit to hit them in the flank and deny steadfast. With only 3-4 attacks, static res should do the rest and pursuit will run them down (hopefully).
White Lions - with S6, the SV cowboy will not work so well and this requires some effort to address. To begin with, sallies are gold even though the WL will get a 4+ AS against any wounds. On average, a couple well placed sallie templates will drop 10+ models so that will cut them down to a manageable size in short order. A double flanking is required to tackle the unit and I find that ranked skinks/skrox are best for this due to speed and mobility. Also, I would rather lose a 5 point skink to S6 than saurus or TG. A combo of steggie on one flank and ranked skinks also works as steggie will likely survive several rounds and his thunderstomp will add active CR while the skinks provide static at minimal cost and with only 3-4 lost to attacks per round while WL are ground down.
Magic - BoWD only works against wounding hits so hexes and buffs should be the focus of the magic phase. Heavens works really well since the spells are low casting values, skinks can cast them and multiple attempts per phase are possible for the signature. Even a single success means -1 to HE to hit and LD. If using high magic, casting unforging at the banner bearer will quickly draw out dispel dice, thereby allowing other spells to get through.
If all else fails, you can always sacrifice and redirect the units with skinks while killing off the rest of the army.