After quite a while Apple has finally resolved the problem with iOS VPN Manager, which stopped DNS Override from properly overwriting preferred DNS in iOS. The issue has been finally resolved in iOS 10 beta 2 after over 4 months of our continuous efforts.
We will be asking Apple to include the fix in upcoming stable iOS 9.3.3 release (currently at public beta 4). This would allow our users to take advantage of DNS Override before public iOS 10 release (planned for Fall 2016).

Update / 13th of September 2016:
Today iOS 10 was released. We can confirm that the problem is now resolved in iOS 10. Unfortunately Apple did not include the fix in any of the iOS 9.3 releases (9.3 – 9.3.5).