I will try in the next few days to do as you indicate. In thinking about the Kruskal-Szekeres coordinates I see potentially a similar issue. The Kruskal-Szekeres coordinates in (t, r, θ, φ) coordinates are
U = sqrt{1 - r/2m}e^{r/4m}cosh(t/4m)
V = sqrt{1 - r/2m}e^{r/4m}sinh(t/4m)
in the exterior region. If I let (t, r) --- > (τξ^a, ξ^b) thse become
U = sqrt{1 - ξ^b/2m}e^{ξ^b/4m}cosh(τξ^a/4m)
V = sqrt{1 - ξ^b/2m}e^{ξ^b/4m}sinh(τξ^a/4m)
and the differential of these are
dU = [(1 - ξ^b/2m)^{1/2}bξ^{b-1}/4m} -
(1 - ξ^b /2m)^{-1/2}]e^{ξ^b/4m}cosh(τξ^a/4m) +
{1 - ξ^b/2m}e^{ξ^b/4m}e^{ξ^b/4m}sinh(τξ^a/4m)(ξ^adτ + aξ^{a-1}dξ)/4m
dV = [(1 - ξ^b/2m)^{1/2}bξ^{b-1}/4m} -
(1 - ξ^b/2m)^{-1/2}]e^{ξ^b/4m}cosh(τξ^a/4m) +
{1 - ξ^b/2m}e^{ξ^b/4m}e^{ξ^b/4m}sinh(τξ^a/4m)(ξ^adτ + aξ^{a-1}dξ)/4m
which are rather complicated. The whole metric is
ds^2 = 32m^3e^{-ξ^b/2m)/(ξ^b)(dU^2 - dV^2) + r^2dΩ^2
with the dU and dV substituted from above.
In looking at this without working out the exact metric coefficients there are some things I can see. The ξ^{-b} counters the ξ^{2b-2} into ξ^{b-2}, the ξ^{2a} into ξ^{2a-2b} and the ξ^{2a-2} into ξ^{2a-2b -1}. These do not diverge if a = 1 and b >= a +1/2. However, there is a term (1 - ξ^b/2m)^{-1/2}, which diverges for ξ^b = 2m. As a result we are left with the horizon blow up.
I'd have to work this out more explicitly (rather than typing in the analysis as I think it) to make sure this is on the mark. We seem to be left with a singular condition at the horizon still.
Cheers LC