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You should have a look at the expression test, wich is a very huge. Hence, when turned numerical, a lot of roundings accumulate. Instead, you could compute the numerical integral directly (the second part is the precision of the result):
sage: numerical_integral(BW,m-20*Gamma,m+20*Gamma)
(1.039353768756414, 7.667179586654595e-10)
As you can see, the integral is larger than 1. If you want more precision, you can use real ball field:
sage: CC = ComplexBallField(1000)
sage: f = fast_callable(BW,vars=[x],domain=CC)
sage: CC.integral(lambda x,_: f(x) ,CC(m-20*Gamma),CC(m+20*Gamma))
[1.0393537687564144005635175005493502562514240251865059997675473690905082074521147152980391279501557766285787071456276287443389069265579835090610752201107443966192623303858945758537114342432084054091316038569497030744623566434274060346642219779086790951559180203731682971294215371002868275171917298204 +/- 4.40e-299]
