It seems the exact formula given by David Bar Moshe in the predecessor to this question,
is the place to start. (To cut and paste: this formula is
Vol(delta-ball) = 2^(m^2)/(pi^m * m!)* int_ phi_ 1<=phi_ 2 <= . . . <=phi_ m <=2 * arcsin(sqrt(delta/2))*
prod_ 1 < = j < k < = m (cos(phi_ k)-cos(phi_ j)^2 prod_ l sin^2(phi_ l) dphi_ 1 . . . dphi_ k.
when n=2m+1, and something very similar for n=2m (my guess is that any lower bound for the former can basically be achieved for the latter).
I am tempted to start dyadically decomposing based on the approximate size of phi_l and of phi_k-phi_j, but this is likely to lose some exponential factors of n.
An alternative is Fourier decomposition; the formula above does suggest that there is a reasonably nice formula for the integral of exp( i theta r ) where r is the distance to the identity and theta is a parameter. If one gets sufficiently good control on that integral, one can then try Fourier inversion.
