Not quite an answer, but:

1. The Kemnitz/Harborth conjecture was proved for cubic planar graphs in:

Straight line embeddings of cubic planar graphs with integer edge lengths Jim Geelen1, Anjie Guo2,†, David McKinnon3 (Journal of Graph Theory, 2008)

They state a condition which would imply Kemnitz/Harborth (property 3.1 in their paper).

They cite the following theorem, which is related to, but not the same as, you conjecture:

Theorem 2.1 (Berry 1992, Acta Arith). If $A, B, C \in \mathbb{R}^2$ are non-collinear points such that $dist(A, B), dist(A, C)^2,$ and $dist(B, C)^2$ are rational, then the set of points that are a rational distance from each of $A, B, C$ forms a dense subset of $\mathbb{R}^2.$

Not quite an answer, but:

1. The Kemnitz/Harborth conjecture was proved for cubic planar graphs in:

Straight line embeddings of cubic planar graphs with integer edge lengths Jim Geelen1, Anjie Guo2,†, David McKinnon3 (Journal of Graph Theory, 2008)

They state a condition which would imply Kemnitz/Harborth (property 3.1 in their paper).

They cite the following theorem, which is related to, but not the same as, what you conjecture:

Theorem 2.1 (Berry 1992, Acta Arith). If $A, B, C \in \mathbb{R}^2$ are non-collinear points such that $dist(A, B), dist(A, C)^2,$ and $dist(B, C)^2$ are rational, then the set of points that are a rational distance from each of $A, B, C$ forms a dense subset of $\mathbb{R}^2.$