the repair and/or maintenance of a certified airplane requires the parts specified for that aircraft by type
The aircraft concerned does not have a standard CofA and is flying under an LAA approved permit to fly. So the authority for approving repairs lies with the LAA either directly or through an authorised inspector.
And, as that permit only authorises flight in UK airspace, it implies that flight authorisation needs to be obtained for each foreign state that is transitted, with any restrictions they choose to apply.
I've no idea to what extent this is being done 'by the book'. One may only guess!