"Something" is wrong in (probably) the red amplification,
or the red gun of the CRT. You'll have to put a 'scope in
the path and see what's fluctuating.

Unfortunately I'm not brave enough to rummage inside a CRT
TV and risk death. Is this likely to be something that's easily
fixed and would it tend to be costly?

It's almost impossible to be electrocuted -- the HV can't deliver enough
current.

Most (but not all) TV repairs are quite simple -- the hard part is figuring
out what's wrong.

Most problems cannot be diagnosed at a distance -- even Dr House has to see
the patient. If I could tell you exactly what's wrong, while sitting at my
computer -- I'd be pulling down big bucks in the service business.

Either find a competent, honest service tech, or buy a new set.