Can a CRI teach IMC/IR(R)

The relevant reference is CAP 804, Section 5 Part E para 3.6 (p 680 of the PDF), Instructors wishing to teach for the IMC/IR(R) Rating

The IFR experience pre-requisites aren’t necessary. See the note at the end of 3.6.3.

An IRI can, however, instruct for the IR without being an FI.

There is a thread on CRI privileges (mostly based a night), where there are diverging optinions (where instruction (e.g. EFIS) happens to be at night / IMC, but not FOR the ratings), but I think everyone would agree with Balliol on this one, the CRI definitely can’t instruct FOR IRR/IR/Night ratings

No they can’t