How to avoid machine underflow for plotting and integrating?

Question

I am trying to integrate the following integrand, which I have plotted:

Manipulate[
 Plot[1/k Evaluate@
     D[Erfc[k] + PolyLog[0, -E^(-(k - \[Mu])/T)] + 
       PolyLog[0, -E^(-(k + \[Mu])/T)], {k, 2}] 5 T Sec[x]^2 /. 
   k -> (\[Mu] + 5 T Tan[x]), {x, -ArcTan[\[Mu]/(5 T)], \[Pi]/2}, 
  Frame -> True, PlotRange -> All, 
  PlotRangePadding -> None], {{\[Mu], 0.5}, 0, 1}, {{T, 0.01}, 0.0001,
   0.1}]

The plot is fine until we slide T to about 0.002 or below, at which point machine underflow errors abound and the plot breaks. In this case, the machine underflow does not appear to be a superficial issue where we should set the value to zero, since the plot completely breaks beyond this point.

I have tried increasing WorkingPrecision and using SetPrecision on various parts of the integrand, but to no avail. Is there a simple way to resolve this?