We're not sure if remi has backported security fixes with 5.4. OLS will work with other PHP sapi, but they won't have the same performance or security (it's going to be a cost of one or the other).
If you are not limited to 5.4 (meaning the PHP code you have doesn't require 5.4), you can yum install one of our more recent lsphp packages and set up OLS to use that. That way, you can keep up to date with all the latest features and security patches (7.0+ is a big improvement over 5.x).
We can always help you with this process if you're willing to make the change and run into any issues!
1. Which repo does the PHP (php-fpm) you mentioned come from? Does it have some more patch than remi? Normally remi has already in the leading position for PHP.
2. LSAPI is the LiteSpeed way to communicate with php, which is also faster than other apache ways. You can compile PHP yourself with patch you need.