How do I rebuild a snapshot without getting a new buildLabel

From the workspace that our build engine runs against, we obviously
have numerous snapshots that have build up over time for our daily
builds. Our underlying code then uses the property values for buildLabel
and buildDefinitionId to display in the logs and trace of the product,
so we know what version people are running.

When I want to rebuild one of those, it seems the only way is to create
a new workspace based off one of those snapshots, and to then request a
personal build using this new workspace. However, that will presumably
create a new snapshot (for the current date etc.), whereas what I want
it to do is to kick off a re-build of the code at that snapshot level
and also using the same value for buildLabel. Is this possible?