This is only a partial answer, but it's too long to fit in the comments.

For any (commutative) ring $R$, there is a canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$, where $\text{Cl}(R)$ is the ideal class group of $R$, that is, the group of all invertible ideals of $R$ modulo its subgroup of principal regular ideals of $R$.

If $R$ is an integral domain or a Noetherian ring, or more generally if $R$ is a ring with few zerodivisors, then the canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$ is an isomorphism.

Also, if $R$ is a regularly Bezout ring, that is, if every finitely generated regular ideal of $R$ is principal, then clearly the ideal class group $\text{Cl}(R)$ of all invertible ideals of $R$ modulo principal ideals of $R$ is trivial. In fact, a regularly Bezout ring is equivalently a Prufer ring with trivial ideal class group. Prufer rings (with or without zerodivisors) are a well-studied class of rings. The proper invariant to study Prufer rings is the ideal class group rather than the Picard group.

It follows from the comments above any regularly Bezout (or Bezout) ring with nontrivial Picard group cannot be a ring with few zerodivisors (and therefore cannot be Noetherian or a domain). So, to find an example, you might start by looking at the construction of such rings in the literature. It seems like one should be able to make many examples that are Bezout, but then finding one with nontrivial Picard group could be hard.

Marot rings are a generalization of rings with few zerodivisors, and there are also many constructions of rings that are not Marot in the literature.

This is only a partial answer, but it's too long to fit in the comments.

For any (commutative) ring $R$, there is a canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$, where $\text{Cl}(R)$ is the ideal class group of $R$, that is, the group of all invertible fractional ideals of $R$ modulo its subgroup of principal regular fractional ideals of $R$.

If $R$ is an integral domain or a Noetherian ring, or more generally if $R$ is a ring with few zerodivisors, then the canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$ is an isomorphism.

Also, if $R$ is a regularly Bezout ring, that is, if every finitely generated regular ideal of $R$ is principal, then clearly the ideal class group $\text{Cl}(R)$ of all invertible ideals of $R$ modulo principal ideals of $R$ is trivial. In fact, a regularly Bezout ring is equivalently a Prufer ring with trivial ideal class group. Prufer rings (with or without zerodivisors) are a well-studied class of rings. The proper invariant to study Prufer rings is the ideal class group rather than the Picard group.

It follows from the comments above any regularly Bezout (or Bezout) ring with nontrivial Picard group cannot be a ring with few zerodivisors (and therefore cannot be Noetherian or a domain). So, to find an example, you might start by looking at the construction of such rings in the literature. It seems like one should be able to make many examples that are Bezout, but then finding one with nontrivial Picard group could be hard.

Marot rings are a generalization of rings with few zerodivisors, and there are also many constructions of rings that are not Marot in the literature.

This is only a partial answer, but it's too long to fit in the comments.

For any (commutative) ring $R$, there is a canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$, where $\text{Cl}(R)$ is the ideal class group of $R$, that is, the group of all invertible ideals of $R$ modulo its subgroup of principal regular ideals of $R$.

If $R$ is an integral domain or a Noetherian ring, or more generally if $R$ is a ring with few zerodivisors, then the canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$ is an isomorphism.

Also, if $R$ is a regularly Bezout ring, that is, if every finitely generated regular ideal of $R$ is principal, then clearly the ideal class group $\text{Cl}(R)$ of all invertible ideals of $R$ modulo principal ideals of $R$ is trivial. In fact, a regularly Bezout ring is equivalently a Prufer ring with trivial ideal class group. Prufer rings (with or without zerodivisors) are a well-studied class of rings. The proper invariant to study them is the ideal class group rather than the Picard group.

It follows from the comments above any regularly Bezout (or Bezout) ring with nontrivial Picard group cannot be a ring with few zerodivisors (and therefore cannot be Noetherian or a domain). So, to find an example, you could start by looking to examples of such rings in the literature.

This is only a partial answer, but it's too long to fit in the comments.

For any (commutative) ring $R$, there is a canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$, where $\text{Cl}(R)$ is the ideal class group of $R$, that is, the group of all invertible ideals of $R$ modulo its subgroup of principal regular ideals of $R$.

If $R$ is an integral domain or a Noetherian ring, or more generally if $R$ is a ring with few zerodivisors, then the canonical inclusion $\text{Cl}(R) \longrightarrow \text{Pic}(R)$ is an isomorphism.

Also, if $R$ is a regularly Bezout ring, that is, if every finitely generated regular ideal of $R$ is principal, then clearly the ideal class group $\text{Cl}(R)$ of all invertible ideals of $R$ modulo principal ideals of $R$ is trivial. In fact, a regularly Bezout ring is equivalently a Prufer ring with trivial ideal class group. Prufer rings (with or without zerodivisors) are a well-studied class of rings. The proper invariant to study Prufer rings is the ideal class group rather than the Picard group.

It follows from the comments above any regularly Bezout (or Bezout) ring with nontrivial Picard group cannot be a ring with few zerodivisors (and therefore cannot be Noetherian or a domain). So, to find an example, you might start by looking at the construction of such rings in the literature. It seems like one should be able to make many examples that are Bezout, but then finding one with nontrivial Picard group could be hard.

Marot rings are a generalization of rings with few zerodivisors, and there are also many constructions of rings that are not Marot in the literature.