All training for a LAPL shall be conducted at an Approved Training Organisation
FCL.115 LAPL — Training course
Applicants for an LAPL shall complete a training course within an approved training organisation. The course shall include theoretical knowledge and flight instruction appropriate to the privileges given.
As I understand it all aircraft used for Approved Training will require an EASA C of A which rules out kit builds.
Re Converting LAPL to a PPL
FCL.210.A PPL(A) - Experience requirements and crediting
(b) Specific requirements for applicants holding an LAPL(A). Applicants for a PPL(A) holding an LAPL(A) shall have completed at least 15 hours of flight time on aeroplanes after the issue of the LAPL(A), of which at least 10 shall be dual flight instruction completed in a training course at an approved training organisation. This training course shall include at least 4 hours of supervised solo flight time, including at least 2 hours of solo cross-country flight time with at least 1 cross-country flight of at least 270 km (150 NM), during which full stop landings at 2 aerodromes different from the aerodrome of departure shall be made.