Like Jason I'd be more inclined to look at this kind of embedding in a geometric or conceptual way. But for your purposes a concrete description seems needed. For this it might be simpler to work in the Lie algebra setting. I don't see immediately what the group viewpoint does for you, since the centerless group $G_2$ embeds in either the special orthogonal group or its slightly elusive simply connected covering group Spin. The basic roots-and-weights technology here depends just on the Lie algebra embedding. For this to be made concrete, however, you'd have to line up the two Cartan subalgebras (Lie algebra of maximal tori) in a compatible way and relate the two root systems via simple roots.

A description of the embedding here is given (in an adapted version going back to one of the Paris seminars) in the first part of section 19.3 of my 1972 Lie algebra textbook. (Needless to say, I haven't spent a lot of time with this material since then.) The main deficiency in the details written down is that a choice of simple roots for $G_2$ isn't made explicit. So you'd have to work that out further. Since the weight lattice and root lattice of $G_2$ coincide, passage to the duals can be done over $\mathbb{Z}$.

At first glance it looks as if the columns of your $3 \times 2$ matrix might be something like $(-2, 1, 1)$ and $(1,-1,0)$. But this may not be quite right, so you'd need to relate the roots and weights of the two Lie algebras (which have respective dimensions 14, 21) more carefully. I'll take a closer look at this if I have time later on.

P.S. I've adjusted my first estimate about the matrix, but in any case the translation from groups to Lie algebras may need some clarification. It's worth looking at the way Bourbaki constructs the $G_2$ root system in the 3-dimensional space also used for their construction of $B_3$. (This is also what I followed in section 9 of my book.) Comparing coweights or Cartan subalgebras then looks natural in this setting. Of course, the two Cartan matrices contain the data determining these constructions.

Like Jason I'd be more inclined to look at this kind of embedding in a geometric or conceptual way. But for your purposes a concrete description seems needed. For this it might be simpler to work in the Lie algebra setting. I don't see immediately what the group viewpoint does for you, since the centerless group $G_2$ embeds in either the special orthogonal group or its slightly elusive simply connected covering group Spin. The basic roots-and-weights technology here depends just on the Lie algebra embedding. For this to be made concrete, however, you'd have to line up the two Cartan subalgebras (Lie algebra of maximal tori) in a compatible way and relate the two root systems via simple roots.

A description of the embedding here is given (in an adapted version going back to one of the Paris seminars) in the first part of section 19.3 of my 1972 Lie algebra textbook. (Needless to say, I haven't spent a lot of time with this material since then.) The main deficiency in the details written down is that a choice of simple roots for $G_2$ isn't made explicit (though it is in Bourbaki). So you'd have to work that out further. Since the weight lattice and root lattice of $G_2$ coincide, passage to the duals can be done over $\mathbb{Z}$.

ADDED: With apologies for the delay in filling in some details, I still have some doubts about the basic setting here. On the level of root lattices it seems fairly clear geometrically. For the embedding $\mathfrak{g}_2 \hookrightarrow \mathfrak{so}_7$, one can work over $\mathbb{R}$ (since the Lie algebras are split). Here the bigger root lattice lives in a 3-dimensional euclidean space with standard orthonormal basis $\varepsilon _1, \varepsilon_2, \varepsilon_3$.

For $B_3$ there are 9 positive roots. Two simple roots are long: $\alpha_1 = \varepsilon_1 - \varepsilon_2$ and $\alpha_2 = \varepsilon_2 - \varepsilon_3$. A third simple root is short: $\alpha_3=\varepsilon_3$. Then the root lattice for $G_2$ lies in the plane through 0 defined by the condition that coordinates sum to 0. Here you can take a short simple root $\alpha = \alpha_1$ and a long simple root $\beta = \varepsilon_2 + \varepsilon_3 -2\varepsilon_1$. Then you can compute your $2 \times 3$ or $3 \times 2$ matrix in terms of these bases, keeping in mind that for $G_2$ the root lattice equals the weight lattice and maps into the root lattice for $B_3$  (which however has index 2 in its weight lattice). My quick arithmetic wasn't reliable, but while the matrix you get depends on the choice of bases, the relationship between the two root systems depends only on the Cartan matrices.

Like Jason I'd be more inclined to look at this kind of embedding in a geometric or conceptual way. But for your purposes a concrete description seems needed. For this it might be simpler to work in the Lie algebra setting. I don't see immediately what the group viewpoint does for you, since the centerless group $G_2$ embeds in either the special orthogonal group or its slightly elusive simply connected covering group Spin. The basic roots-and-weights technology here depends just on the Lie algebra embedding. For this to be made concrete, however, you'd have to line up the two Cartan subalgebras (Lie algebra of maximal tori) in a compatible way and relate the two root systems via simple roots.

A description of the embedding here is given (in an adapted version going back to one of the Paris seminars) in the first part of section 19.3 of my 1972 Lie algebra textbook. (Needless to say, I haven't spent a lot of time with this material since then.) The main deficiency in the details written down is that a choice of simple roots for $G_2$ isn't made explicit. So you'd have to work that out further. Since the weight lattice and root lattice of $G_2$ coincide, passage to the duals can be done over $\mathbb{Z}$.

At first glance it looks as if the columns of your $3 \times 2$ matrix might be something like $(2, -1, -1)$ and $(1,-1,0)$. But this may not be quite right, so you'd need to relate the roots and weights of the two Lie algebras (which have respective dimensions 14, 21) more carefully. I'll take a closer look at this if I have time later on.

Like Jason I'd be more inclined to look at this kind of embedding in a geometric or conceptual way. But for your purposes a concrete description seems needed. For this it might be simpler to work in the Lie algebra setting. I don't see immediately what the group viewpoint does for you, since the centerless group $G_2$ embeds in either the special orthogonal group or its slightly elusive simply connected covering group Spin. The basic roots-and-weights technology here depends just on the Lie algebra embedding. For this to be made concrete, however, you'd have to line up the two Cartan subalgebras (Lie algebra of maximal tori) in a compatible way and relate the two root systems via simple roots.

A description of the embedding here is given (in an adapted version going back to one of the Paris seminars) in the first part of section 19.3 of my 1972 Lie algebra textbook. (Needless to say, I haven't spent a lot of time with this material since then.) The main deficiency in the details written down is that a choice of simple roots for $G_2$ isn't made explicit. So you'd have to work that out further. Since the weight lattice and root lattice of $G_2$ coincide, passage to the duals can be done over $\mathbb{Z}$.

At first glance it looks as if the columns of your $3 \times 2$ matrix might be something like $(-2, 1, 1)$ and $(1,-1,0)$. But this may not be quite right, so you'd need to relate the roots and weights of the two Lie algebras (which have respective dimensions 14, 21) more carefully. I'll take a closer look at this if I have time later on.

P.S. I've adjusted my first estimate about the matrix, but in any case the translation from groups to Lie algebras may need some clarification. It's worth looking at the way Bourbaki constructs the $G_2$ root system in the 3-dimensional space also used for their construction of $B_3$. (This is also what I followed in section 9 of my book.) Comparing coweights or Cartan subalgebras then looks natural in this setting. Of course, the two Cartan matrices contain the data determining these constructions.