= Solution
Introduce three independent local parameters $\alpha_j(x)$. A consistent convention for the three Abelian <gauge transformations> is
$$
\boxed{B_\mu^j\longmapsto B_\mu^j-\frac1g\partial_\mu\alpha_j,\qquad
\phi_1\longmapsto e^{i(\alpha_1-q\alpha_2)}\phi_1,\qquad
\phi_2\longmapsto e^{i(\alpha_2-q\alpha_3)}\phi_2.}
$$
Indeed, $g(B_\mu^k-qB_\mu^{k+1})$ shifts by $-\partial_\mu(\alpha_k-q\alpha_{k+1})$, cancelling the derivative of the scalar phase. Thus $D_\mu\phi_k$ transforms by the same phase as $\phi_k$. The two <kinetic terms>, the absolute-value potentials and all Abelian <field> strengths are invariant.
In units where the gauge <coupling> is $g$, the charge vectors under the three factors are
$$
\boxed{Q(\phi_1)=(1,-q,0),\qquad Q(\phi_2)=(0,1,-q).}
$$
Locally this is the gauge algebra of $U(1)^3$. To interpret it globally as three compact circle groups, the charges must belong to a compatible charge lattice: in the stated unit-charge convention integer $q$ is sufficient, while arbitrary irrational $q$ is not a single-valued compact-group representation. The local transformations and <mass> calculation remain meaningful for general positive real $q$ as a Lie-algebra calculation, or with noncompact Abelian factors.
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