While I have not attempted this type of migration personally; one idea that comes to mind is to copy the file system contents you wish to keep to S3 as an archive.
On LightSail download and expand the archive where you need it on the LightSail instance.
Please see my other comment concerning use of LightSail; TL;DR: LightSail should NOT be used for mission critical commercial applications.
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While I have not attempted this type of migration personally; one idea that comes to mind is to copy the file system contents you wish to keep to S3 as an archive.
On LightSail download and expand the archive where you need it on the LightSail instance.
Please see my other comment concerning use of LightSail; TL;DR: LightSail should NOT be used for mission critical commercial applications.