I have been told this is not possible due to a limitation with Harvest but I am almost certain it worked reliably in Asana.
Is there absolutely no way you can make time entries push into ClickUp regardless of which Harvest account you switch to?
Asana also did quite a good job of intelligently knowing which Harvest project and task to bind to based, I guess, on the task or list name.
ClickUp seems not to remember and it's easy for people to inadvertently clock to the wrong task or project in Harvest.