We could offer WebPhone designed for your needs over modern WebRTC protocol instead of old browser SIP addons like on link you provided
WebRTC support for Asterisk should be enabled (native support from Asterisk 11, for other versions WebRTC2SIP server aplication must be installed)
Benefits:
- customer shouldn't install any additional software (just any modern browser)
- from Asterisk 12 codec Opus could be used (good voice quality with any internet conditions, even on bad mobile connections - it is famous Skype codec), in case of using WebRTC2SIP Opus codec also could be enabled
- easy site-interaction using javascript functions
g729, g711, gsm codecs could be used, but opus codec will be better for your purposes