Found the answer here.
The key bits being:
Const PR_ATTACH_MIME_TAG = "http://schemas.microsoft.com/mapi/proptag/0x370E001E"
Const PR_ATTACH_CONTENT_ID = "http://schemas.microsoft.com/mapi/proptag/0x3712001E"
Const PR_ATTACHMENT_HIDDEN = "http://schemas.microsoft.com/mapi/proptag/0x7FFE000B"
...
Set colAttach = l_Msg.Attachments
For x = 1 To Items.Count
Set l_Attach = colAttach.Add(Items.Item(x))
Set oPA = l_Attach.PropertyAccessor
oPA.SetProperty PR_ATTACH_MIME_TAG, ItemTypes.Item(x)
oPA.SetProperty PR_ATTACH_CONTENT_ID, "item" & x
oPA.SetProperty PR_ATTACHMENT_HIDDEN, True
Next
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