The two papers cited below might be of some interest to you:

Theorem 3.3 of (1) says that for a Krull domain $R$, $Cl(R)/Pic(R)$ is torsion iff local class groups are torsion at the maximal ideals. It is also a remark before Theorem 13 of (2), attributed to Chouinard, that any abelian group appears as the Class group of some local Krull domain. So for constructing an example as you want we may choose a local Krull domain $R$ with $Cl(R)$ a free abelian group and try to see if we can manage the image of $Pic$ in $Cl$ to be of finite index. I am unable to construct such an example right away, however its existence seems morally possible to me.

(1) Anderson, Globalization of some local properties.., MR0652428

(2) Bouvier, The local class group.. , MR0681946

The two papers cited below might be of some interest to you:

Theorem 3.3 of (1) says that for a Krull domain $R$, $Cl(R)/Pic(R)$ is torsion iff local class groups are torsion at the maximal ideals. It is also a remark before Theorem 13 of (2), attributed to Chouinard, that any abelian group appears as the Class group of some local Krull domain. So for constructing an example as you want we may choose a local Krull domain $R$ with $Cl(R)$ a free abelian group and try to see if we can manage the image of $Pic$ in $Cl$ to be of finite index. I am unable to construct such an example right away, however its existence seems morally possible to me.

(1) Anderson, Globalization of some local properties.., MR0652428

(2) Bouvier, The local class group.. , MR0681946

The two papers referred below might be of some interest to you:

Theorem 3.3 of (1) says that for a Krull domain $R$, $Cl(R)/Pic(R)$ is torsion iff local class groups are torsion at the maximal ideals. It is also a remark before Theorem 13 of (2), attributed to Chouinard, that any abelian group appears as the Class group of some local Krull domain. So for constructing an example as you want we may choose a local Krull domain $R$ with $Cl(R)$ a free abelian group and try to see if we can manage the image of $Pic$ in $Cl$ to be of finite index. I am unable to construct such an example right away, however its existence seems morally possible to me.

(1) Anderson, Globalization of some local properties.., MR0652428

(2) Bouvier, The local class group.. , MR0681946

The two papers cited below might be of some interest to you:

Theorem 3.3 of (1) says that for a Krull domain $R$, $Cl(R)/Pic(R)$ is torsion iff local class groups are torsion at the maximal ideals. It is also a remark before Theorem 13 of (2), attributed to Chouinard, that any abelian group appears as the Class group of some local Krull domain. So for constructing an example as you want we may choose a local Krull domain $R$ with $Cl(R)$ a free abelian group and try to see if we can manage the image of $Pic$ in $Cl$ to be of finite index. I am unable to construct such an example right away, however its existence seems morally possible to me.

(1) Anderson, Globalization of some local properties.., MR0652428

(2) Bouvier, The local class group.. , MR0681946