So I did some digging, and right now all I can conclude is that the AIM-7R physically existed and final prototypes were tested, flight performance is likely to be identical to AIM-7M, but I can’t find details on its seeker.
“APPROVED NAVY TRAINING SYSTEM PLAN FOR THE AIM-7M/P SPARROW MISSILE SYSTEM N88-NTSP-A-50-8008C/A MAY 1998” says that
“Training requirements for the AIM/RIM-7R were removed because the program was suspended following completion of its Operational Evaluation (OPEVAL).”
and
“The AIM/RIM-7R was the latest Sparrow new development, but the program was halted in the first quarter of FY97 following completion of its DT/OT program.”
This indicates that finalised prototypes of the AIM-7R underwent testing, so performance data should exist somewhere (that somewhere doesn’t seem to be the internet).
Additionally, Force 2001, A Program Guide to the U.S. Navy and Uses of DARPA Materials Sciences Technology in DoD Systems. and many other sources all indicate that the AIM-7R seeker (or at least the IR component) was part of the Missile Homing Improvement Program, which was also applied to the SM-2 Block IIIB, which did enter service, so if data exists on that we can assume it to be the same as the AIM-7R (however, it seems that the SM-2 Block IIIB’s IR seeker is mounted differently).
Artillery, Missiles & Military Transport (20th Century Military Series) by Christopher Chant goes into a bit more detail. It says that the IR seeker is derived from the Sidewinder (model not specified).
According to Chant, the IR seeker is activated first and attempts to acquire a lock. If IR lock is not achieved, it goes into SARH mode until IR lock is acquired.
Finally, pictures of the seeker:


These come from this site: http://sistemasdearmas.com.br/aam/aim7sparrow.html

And this comes from here: View topic - "Dual mode" Sparrow? • F-16.net. Apparently it’s from a book but I can’t find it online. It’s also mislabeled as a AIM-7P which definitely was not dual mode.
Basically the tiny window at the tip is supposed to be the IR seeker.
Then there’re these two photos:
But this seeker is much larger than the previous images, so I’m not sure if this is a different prototype, or an entirely different missile. Alternatively, it could be that the earlier images are wrong and this is the actual AIM-7R. These two photos actually seem a bit more credible since these are mounted on a F-4 for testing while the prior ones seem like models, and if the seeker is adapted from the Sidewinder this does seem a bit more true to the dimensions, but this arrangement seems like it would substantially disrupt the SARH guidance, so maybe this is a purely IR version of the Sparrow?