One point of possible confusion just occurred to me and it doesn't appear to have been visited explicitly in the preceding posts.
If folk are looking at the barberpole, then the limiting crossover will occur at the same level for a given Type as Vmo and Mmo will be the constant TC limits for the Type at the crossover level.
That is to say, the barber pole will follow Vmo on the intial climb - note that, for some Types, Vmo may have a few kinks as the climb progresses - and, when the level causes the Mmo to equal Vmo, crossover occurs. Same each flight.
However, most operators don't fly the climb at barberpole. Therefore, the operator's IAS/IMN schedule will determine crossover quite independent of whatever the barberpole may be doing.