Hey @Jodi_Williams,
yeah exactly you won‘t be able to export comments directly via Asana.
Maybe this helps: Export All Data from Asana - #9 by Pawel_Swiderski
And this integration seems to offer more export options as well.
Now regarding what plan your client should have it really depends how you had set up the project for him. Like for example if custom fields were set up then he would definitely need the premium plan and if you had rules also then business.
Now maybe he would not need to upgrade the organization but only a team.
Maybe there is another way to set up a separate team in your organization and then transfer it to the client. I recommend to contact Asana support to see if there are other options 