I bit the bullet, crossed my fingers, and gave it a try, and it seems to be working. I made a list of all the REMI PHP RPMs I had installed (rpm -qa | grep php | grep remi), removed them all (rpm -e --nodeps php php-common ...), installed the PHP RPMs from the distro ISO to make sure I got them all (yum install php-*), yum update, signal-event post-upgrade, signal-event reboot.
I then installed the software collections php54 and php55 and made the appropriate settings for the owncloud and joomla ibays. They're working as well. Thanks for the input!