Values at non-positive integers of partially twisted multiple zeta-functions II

We study the values at non-positive integer points of multi-variable twisted multiple zeta-functions, whose each factor of the denominator is given by polynomials. The fully twisted case was already answered by de Crisenoy. On the partially twisted case, in one of our former article we studied the case when each factor of the denominator is given by linear forms or power-sum forms. In the present paper we treat the case of general polynomial denominators, and obtain explicit forms of the values at non-positive integer points. Our strategy is to reduce to the theorem of de Crisenoy for the fully twisted case, via the multiple Mellin-Barnes integral formula. We observe that in some cases the obtained values are transcendental.