In short:
NMSA 66-7-9 grants local authorities broad powers to regulate bicycles. They can require you to use the sidewalk, not use the sidewalk, use a sidepath, or completely ban you from certain streets. Establishing local laws for stop signs and stop lights for bicyclists is within their power.
Albuquerque has such an ordinance.
§ 8-3-3-5 OBEDIENCE TO TRAFFIC CONTROL DEVICES.
(A) Any
person operating a bicycle shall obey the instructions of official
traffic control signals, signs, and other control devices applicable to
vehicles, unless otherwise directed by a police officer.
It doesn't say anything about being exempted from stopping at stop signs or from waiting at a red light until a green is shown. These are the clear meanings of stop signs and control signals in the MUTCD. It's a pretty straightforward argument that the new State law is superseded by this explicit local ordinance regulating bicycles.
It appears that if you perform an Idaho Stop in Albuquerque, you could be stopped and cited.
Any attorneys on this list versed in traffic law who would care to weigh in on this?