With Local 3.3.X there used to be a manual way that you could define SSL, .conf and hosts file entry so that a multisite could handle domain mapping. See https://localwp.com/community/t/local-router-error-message-on-multi-network-instance/1913/13 for details.
With Lightning, whenever the main site is restarted, the custom .conf added to ~\AppData\Roaming\Local\run\router are deleted.
At minimum, do not delete the custom .conf. More preferably, add a way to define related domains within the UI so that ssl, .conf and hosts entry are managed along with the main site by Lightning.