Symmetry does the trick!
We have a base and an acid - lets call them HA and BOH.
Ka = [A][H]/[HA]
Kb = [ B][OH]/[BOH]
[H] = Ka[HA]/0.2
Why 0.2? Because the hydrolysis doesn't go too far and we can safely assume that the dissociated form concentration is almost the same as the concentration of salt. Besides, that's only assumption - we can later check if it was OK, and if not, start again.
Now, first fancy move:
[H][OH] = Ka[HA]/0.2 * Kb[BOH]/0.2 = Kw
[HA][BOH] = 0.2^2*Kw/(Ka*Kb)
And now... SYMMETRY DOES THE TRICK!
Ka and Kb have very similar values, so lets assume that [HA] = [BOH]. If so
[HA]^2 = 0.2^2*Kw/(Ka*Kb)
and [HA] = 0.00022
Getting back to [H] and acid dissociation constant:
[H] = Ka [HA] / 0.2 = 10^(-4.75) * 0.00022 / 0.2 = 1.96E-8
and pH = 7.71
That's the same value as the one calculated with BATE. To be OK we now should check if our assumption was right, putting known values of [H], [OH], [HA] and [BOH] into all equations and checking if they hold - but I can assure you they do.
OK, I am going before my wife filles for divorce....