You can try the following, leave laravel's original htaccess alone however.
Add another .htaccess with only this to base root like lms
RewriteEngine On
RewriteRule ^(.*)$ public/
RewriteRule ^ server.php
That will serve public, and in public you should have this .htaccess:
<IfModule mod_rewrite.c>
<IfModule mod_negotiation.c>
Options -MultiViews -Indexes
</IfModule>
RewriteEngine On
RewriteBase /yoursite/ I only add this line try /lms/
# Handle Authorization Header
RewriteCond %{HTTP:Authorization} .
RewriteRule .* - [E=HTTP_AUTHORIZATION:%{HTTP:Authorization}]
# Redirect Trailing Slashes If Not A Folder...
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_URI} (.+)/$
RewriteRule ^ %1 [L,R=301]
# Send Requests To Front Controller...
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^ index.php [L]
# RewriteRule ^(.*)$ public/ [L]
</IfModule>
I like adding the RewriteBase /yoursite/ so I don't need all the extra slashes in routing and redirecting.
Of course yoursite gets changed to actual url root of site.