FCL.210.A(b) requires that the training course for LAPL(A) to PPL(A) upgrade is completed in a training course at an ATO. All ATOs require approval for Part-FCL training courses - does your 'friendly ATO' have such approval for LAPL to PPL training?
FCL.905.CRI(a) defines the privileges of a CRI. Para (1) makes it clear that these are for 'the issue, revalidation or renewal of a class or type rating for non-complex non-high performance single-pilot aeroplanes, when the privileges sought by the applicant are to fly in single-pilot operations;'
I therefore conclude that the LAPL(A) to PPL(A) course may not be conducted by a CRI - flight instruction for PPL(A) initial issue must be conducted by a Part-FCL FI(A).
The rest of €uroland seems rather less keen on CRIs for SEP work than has been the case in the UK hitherto.