Problem with NIntegrate over a piecewise function

If your second numerical integration uses appropriate options nearly the same results are obtained.

(Those options can be figured out from the messages NIntegrate issues while evaluating your second integral.)

One such way is:

NIntegrate[ff, {y, 0, 1}, {x, y, 1}, 
 Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 10^6}, 
 MaxRecursion -> 1000]

Here is another:

NIntegrate[ff, {y, 0, 1}, {x, y, 1}, MinRecursion -> 3]

(See res3 and res4 in the attached screenshot.)

As I mentioned in a comment, NIntegrate does solve the condition 1.1 x^0.045 < 1 for the singularity at x == b2bar and this causes a problem with the integration, which is itself an issue. But that issue can be avoided by reducing the condition to something NIntegrate can handle. If we throw in the domain restriction 0 <= x <= 1 && 0 <= y <= 1 from the integral, or just the x component 0 <= x <= 1, then Reduce will solve the condition.

ff = PiecewiseExpand[D[Min[G2, 1], x]*x*q2,
  Method -> {"ConditionSimplifier" -> (Quiet[
        Reduce[# && 0 <= x <= 1, x, Reals], Reduce::ratnz] &)}]
NIntegrate[ff, {y, 0, 1}, {x, y, 1}]