The OP asks for examples of groups with solvable word problem for which the intersection problem is undecidable. Such examples certainly exist. Here's one way to construct one.

Let $G$ be a finitely presented, torsion-free group with undecidable word problem. The Rips construction provides a short exact sequence

$1\to K\to\Gamma\to G\to 1$

where $\Gamma$ is torsion-free hyperbolic (in particular, fp, with solvable word problem and many other nice properties) and $K$ finitely generated. In particular, any cyclic subgroup $\langle w\rangle$ intersects $K$ non-trivially if and only if $w\in K$, which is undecidable.

With a little more work, one can combine this idea with the Mihailove construction mentioned by Benjamin Steinberg in his answer to provide counterexamples in which the subgroup $K$ is finitely presented (this time inside a product of hyperbolic groups). This uses the 1-2-3 theorem of Bridson--Baumslah--Miller and Short.

The above techniques are a standard machine for producing finitely generated and presented examples of pathological subgroups of otherwise fairly well behaved groups. For example, you can arrange for $\Gamma$ to be linear over $\mathbb{Z}$ etc.

The OP asks for examples of groups with solvable word problem for which the intersection problem is undecidable. Such examples certainly exist. Here's one way to construct one.

Let $G$ be a finitely presented, torsion-free group with undecidable word problem. The Rips construction provides a short exact sequence

$1\to K\to\Gamma\to G\to 1$

where $\Gamma$ is torsion-free hyperbolic (in particular, fp, with solvable word problem and many other nice properties) and $K$ finitely generated. In particular, any cyclic subgroup $\langle w\rangle$ intersects $K$ non-trivially if and only if $w\in K$, which is undecidable.

With a little more work, one can combine this idea with the Mihailove construction mentioned by Benjamin Steinberg in his answer to provide counterexamples in which the subgroup $K$ is finitely presented (this time inside a product of hyperbolic groups). This uses the 1-2-3 theorem of Baumslag--Bridson--Miller--Short.

The above techniques are a standard machine for producing finitely generated and presented examples of pathological subgroups of otherwise fairly well behaved groups. For example, you can arrange for $\Gamma$ to be linear over $\mathbb{Z}$ etc.