Let $\chi$ be a one-dimensional geometric (in the sense of FM) $p$-adic Galois representation of $G_K$ and let $\psi$ be the Hecke character of $K$ associated to $\chi$ by class field theory. The fact that $\chi$ is de Rham (=pst) at all primes above $p$ imples that $\psi$ is an algebraic Hecke character. Generally, the only algebraic Hecke characters of $K$ are of the form $(\text{finite order})\cdot\mathcal{N}^n$ where $\mathcal{N}$ is the norm character. Under class field theory, $\mathcal{N}$ corresponds to the cyclotomic character, so it comes from geometry; additionally, any finite order character comes from geometry (it arises as the subquotient of the $H^0$ of a zero-dimensional variety). The only time there are more algebraic Hecke characters is when $K$ contains a CM field. Denoting $K_{CM}$ the maximal CM field in $K$, every algebraic Hecke character of $K$ is of the form $(\text{finite order})\cdot(\psi_{CM}\circ\mathcal{N}$ $_{K/K_{CM}})$ where $\psi_{CM}$ is an algebraic Hecke character of $K_{CM}$ and $\mathcal{N}$ $_{K/K_{CM}}$ is the norm from $K$ to $K_{CM}$. Again, finite order characters come from geometry, so this case is reduced to the CM case. As you've mentioned the CM case has been dealt with, so Fontaine–Mazur is true for $\mathrm{GL}(1)$.

Let $\chi$ be a one-dimensional geometric (in the sense of FM) $p$-adic Galois representation of $G_K$ and let $\psi$ be the Hecke character of $K$ associated to $\chi$ by class field theory. The fact that $\chi$ is de Rham (=pst) at all primes above $p$ imples that $\psi$ is an algebraic Hecke character. Generally, the only algebraic Hecke characters of $K$ are of the form $(\text{finite order})\cdot\mathcal{N}^n$ where $\mathcal{N}$ is the norm character. Under class field theory, $\mathcal{N}$ corresponds to the cyclotomic character, so it comes from geometry; additionally, any finite order character comes from geometry (it arises as the subquotient of the $H^0$ of a zero-dimensional variety). The only time there are more algebraic Hecke characters is when $K$ contains a CM field. Denoting $L$ the maximal CM field in $K$, every algebraic Hecke character of $K$ is of the form $(\text{finite order})\cdot(\psi_L\circ\mathcal{N}_{K/L})$ where $\psi_L$ is an algebraic Hecke character of $L$ and $\mathcal{N}_{K/L}$ is the norm from $K$ to $L$. Again, finite order characters come from geometry, so this case is reduced to the CM case. As you've mentioned the CM case has been dealt with, so Fontaine–Mazur is true for $\mathrm{GL}(1)$.