Your integrals have a closed symbolic form, and it seems that the value given by NIntegrate is correct:
In[1742]:=
Integrate[f0[s] Vbar[s]*f0[u] Vbar[u], {s, 0, T}, {u, 0, s}] + \[Rho]*
Integrate[
Vbar[s]*Subscript[\[Lambda], 2][s, u] Vbar[u], {s, 0, T}, {u, 0, s}]
% // N
Out[1742]= (-287 + 3588 E^2 + 7943 E^4)^2/(131072000000 E^8) - (
3 (-962033 + 7920336 E^2 + 1970629 E^4))/(26214400000 E^4)
Out[1743]= 0.000195151
