On the next upkeep Enduring Ideal should be cast again.
Is Enduring Ideal countered by Dovescape, or can I search for other enchantments?
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No, a copy of Enduring ideal won't be countered by Dovescape because the copy isn't cast.
The reminder text for Epic says:
At the beginning of each of your upkeeps, copy this spell except for its epic ability.
However, copies are not cast and thus are not affected by Dovescape:
706.10. To copy a spell, activated ability, or triggered ability means to put a copy of it onto the stack; a copy of a spell isn’t cast and a copy of an activated ability isn’t activated. [...]