BPS 403.php and sucuri

What you want to eliminate first is that a parent htaccess file is actually the root cause of the problem. See this forum topic link and create a child site RewriteRule (exclude/skip) rule in the parent .htaccess file for the elearning child site: http://forum.ait-pro.com/forums/topic/htaccess-files-for-multiple-website-domains/

Example:

# WP REWRITE LOOP START
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]

# Do not apply rules to other child websites &
# do not log errors for these child sites
RewriteRule ^elearning/ - [L]

In the hosting account root htaccess file code change the code to this below by removing the RewriteRule ^successfulflyfishing/elearning/ - [L] RewriteRule. all the other code you posted is good.

# Do not apply rules to other child websites &
# do not log errors for these child sites
RewriteRule ^successfulflyfishing/ - [L]

Ok so I am not seeing any errors right now when using the Sucuri Sitecheck scanner so I assume you still have Security Logging/ErrorDocument commented out. The only common problem that I am aware of with ErrorDocument Security Logging/redirects is when you are using mod_security and in that case you probably want to allow mod_security to handle error logging or most likely you will end up with redirect looping problems or you would disable mod_security and allow BPS to handle error logging instead. Are you using mod_security?

Yes. You are correct. I am not aware of any other ErrorDocument issues/problems besides the known one with mod_security and BPS Security Logging trying to do the same thing – error logging.

Headers check using GET result is a 500 error:

HTTP Status Code: 500 Internal Server Error

Headers: Array
(
    [date] => Thu, 09 Apr 2015 00:49:21 GMT
    [server] => Apache
    [expires] => Thu, 19 Nov 1981 08:52:00 GMT
    [cache-control] => no-store, no-cache, must-revalidate, post-check=0, pre-check=0
    [pragma] => no-cache
    [set-cookie] => PHPSESSID=lmghcvb5t5jj13ae1q5om8o4a0; path=/
    [vary] => Accept-Encoding
    [content-encoding] => gzip
    [content-length] => 702
    [connection] => close
    [content-type] => text/html
)

Logically I believe you have invalid/bad custom code somewhere.
Do these troubleshooting steps:
http://forum.ait-pro.com/forums/topic/read-me-first-free/#bps-free-general-troubleshooting

Note: After doing each troubleshooting step, test whatever is not working to see if it is now working. It could also be possible that 2 things are causing a problem. Example scenario: Doing step 2 and step 3 allow whatever was not working to start working. That would mean both the root .htaccess file and the wp-admin .htaccess file are blocking something legitimate.

1. Optional (not necessary): Make a backup of your .htaccess on the htaccess Core >>> Backup & Restore tab page.
2. On the Security Modes page, select the Deactivate Root Folder BulletProof Mode (Default Mode) Radio button and click the Activate|Deactivate button.
See Custom Code Note
3. On the Security Modes page, select the Deactivate wp-admin Folder BulletProof Mode Radio button and click the Activate|Deactivate button.
See Custom Code Note
4. If an issue/problem is related to Login Security turn Off Login Security on the Login Security & Monitoring page.

Custom Code Note: If you have isolated a problem to the root or wp-admin .htaccess file and you have added additional custom .htaccess code or additional .htaccess code from another plugin to BPS Custom Code then cut (not Copy) all of your additional custom .htaccess code out of all BPS Custom Code text boxes and save that custom .htaccess code to a Notepad or Notepad++ text file, Click the Save Root Custom Code button (or the wp-admin Custom Code button), click the AutoMagic buttons on the Security Modes page and activate BulletProof Modes. You can then further isolate which custom .htaccess code is the problem by adding only 1 block of additional custom code back to a BPS Custom Code text box at a time.

Also check your php error log and server log for clues.

Ok but these things should be fairly quick to do.
Reinstalling WordPress takes about 1 minute >>> Dashboard >>> Updates >>> Re-Install Now button. I took a look at the site. You should eliminate the theme by switching to another theme and back again. You should eliminate this plugin – wp courseware. There are reasons why I think doing these things is worth trying and I won’t bore you with the technical logic of doing this things. Intermittent problems typically indicate a caching problem. Cookie problems can also be caused by a caching problem. I do not see a caching plugin installed on this site based on the Source Code.