Can we always reduce the weights of a weighted graph to rationals and preserve equality relationships?
Identify $W$ with the set of all weights. Let $B$ be a basis of $W$ over the rationals. That is, you can write every weight in $w$ as a linear combination of elements of $B$ with rational coefficients. You can identify the weights with vectors $\vec{w}$ of length $|B|$. Each linear dependency that you want to avoid can be thus encoded as an inequality
$$
\sum_i \alpha_i b_i \neq 0,
$$
where the coefficients $\alpha_i$ are not all zero. We can moreover assume that the $\alpha_i$ are integers (by taking a common denominator).

Let $M$ be an integer larger than all $|\alpha_i|$ appearing in an inequality you are interested in. If we replace $b_i$ with $M^i$ then all the requisite inequalities will be satisfied, while all equalities will also be preserved.