iOS custom keyboard clicks sound
Simple answer: No. You're stuck with the sounds in the interface Apple provides, which have default settings for ringtone, text tone, new voicemail, new mail, sent mail, calendar alerts, reminder alerts and airdrop.

You can customize entries in Contacts to use alternate sounds and ringtones for texts and calls, but that's it.

It may be possible to change the click sound if you jailbreak your iPhone, but I'm just guessing.