SECTION 2
Specific requirements for the flight instructor — FI
FCL.905.FI FI — Privileges and conditions
The privileges of an FI are to conduct flight instruction for the issue, revalidation or renewal of:
(a) a PPL, SPL, BPL and LAPL in the appropriate aircraft category;

So yes - but only in the LAPL category.

FCL.915.FI(b)(2) specifically requires for the applicant of an FI certificate to hold at least a PPL license, so the answer would be no. However the question refers to a CRI certificate. In this case nothing in the FCL.915.CRI prevents a LAPL holder to apply for a CRI certificate

The privileges of a LAPL(A) are listed in FCL.105.A and do not include instruction. A LAPL(A) holder cannot include a CRI(A) certificate in their licence; the minimum requirement would be a PPL(A) with a Part-MED Class 2 Medical Certificate.
So NO!

Instructor privileges are not generally included in the "Privileges" section for the pilot licenses (FCL.205.A for PPL and FCL.305 for CPL license). For that reason the lack of mention of instructor privileges in FCL.105.A cannot be read as not allowing a LAPL(A) pilot to perform instruction as a CRI. So I have to agree with mr. Sabatino. The answer is yes.

FCL.205.A(b) allows for flight instruction by PPL(A) holders, FCL.105.A does not include a similar clause for LAPL(A) holders.
The answer is still NO!!

I do not agree. FCL.205.A(b) is about flight for remuneration, stating an exemption regarding flight instruction. It does not specify whether instruction is within the privileges of a PPL or not. FCL.105.A does not have a similar clause, simply because flight instruction does not constitute an exemption from the not-for-remuneration rule for LAPL holders.
Similarly, the lack of mention of instructor privileges in FCL.305 does not prohibit a CPL holder from instructing.

A LAPL(A) holder does not meet FCL.915(b)(2) because he/she does not hold a Class Rating.
Anyway, why not check what the Danish CAA has to say on the matter?

Hello Nick,

forgive me .. but a LAPL license if it has the SEP (land) rating ... how do you rate it? Certainly not to do the FI course ....

A LAPL(A) does not contain an SEP (Land) Class Rating, it includes some privileges to operate SEP and/or TMG aircraft. There is a legal difference; a CRI(A) must hold an Aeroplane Class Rating in his/her licence and a LAPL(A) does not meet this criterion.