I think you should interpret the coefficient as is.

Notice that your slope in your original model (the coef multiplied to B) is the effect of changing your value of A from some value say A0 to (A0 + 1)base - 1. It would be nice if base of log were 10 then the slope is the effect of increasing A ten fold and subtracting 9.

There is no need to alter the standard error from your original model.

its estimate is asymptotically (big n) normal, no matter how we interpret it. The standard errors are true (asymptotically).