Fed #6s ,even the FC would shoot better and even the Fiocchi #7.5 nickel plated lead should give an ample pattern at 30 yards.
The Fed HW #7 load or the H-13 #7 load in HTL will be a much better pattern than the lead 5s.
What looks bad in a 20 ga at 40 may look pretty sweet at 30 and what looks horrible at 30 may look OK at 25 and yes even 5 yards does make a difference while 10 yds makes a huge difference.
I would next look at another choke if these shells don't work. IF Win comes out with a LB version in 20 gauge with #5 and #6 shot next year a lot of chokes will do if I am assuming correctly. Look at how many different chokes and exit diameters are working for this shell in 12 gauge.