I would ask for as much turnout as possible, even with being ridden twice a day. My thought is that a horse has “his time” while unconfined, able to move freely around at his discretion. He can make choices, not always under control of others. Not just an exercise thing, but a mental break time.
If he has previously been used to turnout, he might actually need that change in his setting.
My Mare went for a “tune-up” last fall. They had no turnout. She was worked once or twice daily, long tiring works. She did her work, but by the end of 5 weeks started getting crabby. I got a call about her, visited to see her acting poorly and said I would be back to pick her up the next week. She had just reached her limit of stall living, needed to come home for regular, daily pasture turn out again. She was back to her pleasant self after a couple days running like a wild Indian!
She just could not tolerate the confinement with no turnout. She got tired in work, but no chance to ever “goof off” alone, make her own choices of walk or run? We used her at home, no unpleasant behaviour shown again. Now we know she does not do well with long times of confinement.
I am sure it was no other problem at the trainer’s. Great feed, big box stalls, groomed daily, worked out of the arena so she saw lots of new things, not bored. I was pretty happy with the training progress she showed us. She just needed “free time” to run and play outside.