Is it possible to make a entity reference field translatable and keep content in existing translations?
Normally you don't need to make the entity reference field translatable to translate the content. This is only necessary if you want to define for each language different target ids.

But this is a different issue. If you enable translation for a field the translated fields have not yet content. This is not specific to reference fields, it is the same for all field types. Only when you create new translations the field values are copied, so that the form is prefilled with the untranslated values. So if you change the field structure in this way and you want to start with all existing translations prefilled you may need an update script for the existing field data.