I didn't recall this detail from the books -- though I do recall the multiple Idaho gholas -- but this entry in the Dune Wiki indicates that the mentat order was outlawed during the reign of Leto II as part of guiding humanity along the Golden Path. What was the reasoning involved here? Is there any justification given in the text for this decision? (In passing, I know the mentat order developed in the aftermath of the Butlerian Jihad, to permit complex calculations/data storage/"expert systems" to continue to exist in the absence of machine intelligences. This leads to a secondary question, that actually motivated the primary one: how could humanity function at the level of development it had achieved with neither artifical intelligences or their human equivalents, the mentat order?)
In short, Leto II banned Mentats in order to deliberately destabilize his rule so that the Mentat abilities would be used to help bring about The Scattering, in accordance with The Golden Path.
The goal of Leto's Golden Path was to "teach humanity a lesson that they will remember in their bones". He created a stable, but deliberately stagnant, rule over humanity that lasted for thousands of years in order to build up the internal pressures of dissent and dissatisfaction that would shatter, and consequently scatter, humanity upon his death.
It was only through this scattering that he foresaw that the Golden Path could be realized, whereby humanity would survive.
The powerful applied logic and memory that Mentats employed could possibly have been harnessed to help further stabilize Leto's rule, but that would have been counter to The Golden Path. He needed fundamental, but hidden, instabilities so that the Scattering would happen once he died. By outlawing the Mentats, he turned them into one of the sources of instability that would ultimately cause his reign to shatter in just the right way.
It is clear that Leto knew that the Bene Gesserit were employing Mentat training, yet allowed it:
"I have spent some time with the Reverend Mother Anteac," he said. "Although she keeps it well hidden, I'm sure she is a Mentat."
"Yes. The Bene Gesserit were bound to disobey me sometime. This form of disobedience amuses me."
"Then you will not punish them?"
"Moneo, I am ultimately the only parent my people have. A parent must be generous as well as severe."
However, this is the only direct mention of Leto's awareness of the Mentats that exist among the Bene Gesserit. The only other mentions in God Emperor of Dune are the confirmation that Leto has banned Mentats, and confirmation that Anteac has been trained as a Mentat.
Regarding your secondary question, which directly ties into your primary question, humanity survived in stagnation without Mentats or thinking machines, just as Leto planned. The lack of widespread higher calculations, logic, and memory prevented humanity from advancing, while Leto's Deocracatic government by way of his Fish Speakers prevented humanity from regressing, created a near-perfect stability (and therefore deliberate stagnation). One of the primary examples of this was Leto's deliberate crippling of the Spacing Guild through his monopoly on the Spice Melange.
Since Guild Navigators rely upon massive doses of Spice not only to generate the prescient visions necessary to navigate space safely, but also to survive due to their massive Spice addictions, Leto was able to force them to obey his rules, and drastically curtail interstellar travel (which, in turn, perpetuates the desired stagnation of humanity). Yet he allowed them just enough Spice to maintain their existence in a capacity that would support the coming Scattering.
Off the cuff - without returning to the source material - I got the impression that mentat 'projection' (their way of calculating possibilities) is too close to prescience for Leto's comfort, and we know any prescient entity can distort the vision of any other (this is how Paul hid his identity from the Guild, for instance - and they return the tactic in Children of Dune).
On the wiki - the mentat article indicates that they were in opposition to the God Emperor (Leto II) and were banished in the scattering and their schools outlawed.
Unfortunately, the wiki doesn't do a very good job of saying exactly where in the canon it sources its information from.