There is a difference between the training for LAPL and the EASA PPL
The only significant difference is that the LAPL is an abreviated version of the PPL course, and possibly not acheivable in the minimum time frame.
It is possible for a LAPL to complete some of the training and then be completed by an EASA instructor.
Any EASA FI may train for the LAPL, but some EASA FIs are limited to teaching for the LAPL only (Those who do not meet the CPL knowledge criteria) If a student is trained by a FI limited to giving instruction for the LAPL, then all of the LAPL training must be completed before it can be recognised towards a PPL.
I think it should so that those looking to progress have the option do have the advantage of the knowledge of the EASA PPL instructor throughout their course.
Both the EASA FI with and without the LAPL limitation have completed exactly the same FI Course and have been examined to the same standard. The only difference is that one has not completed the CPL TK Exams which probably has very little impact on what they would teach in the first 10 hours. You may recall the many BCPL (Restricted) FIs who formed the backbone of the training industry in the 90s; they did not have CPL TK knowledge either!
EASA has allowed the training of FIs who have this limitation simply because they do not meet one ICAO criteria; they are all trained and tested to the same standard, so should be capable of doing exactly the same job.
As a school you need to be aware that the CAA could refuse to issue a licence if some of the training was conducted by an FI who did not have the privilege to give it. The student would be entitled to request their money back.