I was able to compute an expression for the remaining constant in B(e) by expanding the integral for the difference in time as a linear function of the eccentricity near e = 0.
The constant is the derivative of the integral w.r.t. e at e = 0.
The formula gives a good approximation for the time it takes to go from the Vernal Equinox to the Autumnal Equinox if the orbital period, T0 (tropical year), the eccentricity, e, and the angle, θ, are known. There are higher order terms but the error is about 30 minutes or less.