Let $G$ be a topological group. We know that if $G$ is locally compact, then there exists a left-invariant measure on $G$ which assigns finite measure to compact sets (a.k.a. the Haar measure).
Analogously, the Birkhoff-Kakutani theorem says that if $G$ has a countable base at the identity (and therefore everywhere), then there exists a left-invariant metric on $G$ which generates the same topology.
My question is: if $G$ both has a countable base and is locally compact, do we know anything about how this metric relates to the Haar measure? For example, are there conditions under which the closed unit ball (with respect to the metric) is compact (and therefore assigned finite volume)? Or conversely, are there conditions under which we know that a set which has finite measure also has finite diameter?