Will Enduring Ideal and Dovescape played together leave Enduring Ideal countered?
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. [...]