For a loose metric $d$ as above, we can consider the function $$d_1(x,y):=\sup\{|d(x,z)-d(y,z)|;z\in X\}.$$

It is easy to verify that $d_1$ is a metric, and $d(x,y)\leq d_1(x,y)\leq\rho(d(x,y))$ for all $x,y$, so $d_1$ and $d$ generate the same topology.

For a loose metric $d$ as above, we can consider the function $$d_1(x,y):=\sup\{|d(x,z)-d(y,z)|;z\in X\}.$$

It is easy to verify that $d_1$ is a metric, and $d(x,y)\leq d_1(x,y)\leq\rho(d(x,y))$ for all $x,y$, so $d_1$ and $d$ generate the same topology.

Edit: As mentioned in the comments, we can let $d_2=\min(d_1,1)$ if we prefer metrics which don't take the value $\infty$.