someone i was working with build a very successful SaaS with freehand PHP without a framework, but it was clunky, hard to maintain and kept getting hacked.
they hired a 3rd party to rebuild the service from scratch using laravel, unfortunately the project wasn't managed at all and expectations weren't aligned so after about 12 months and a lot of money wasted the rewrite was scrapped. The biggest problem with this approach was they used the waterfall model and expected everything to come together nicely at the end rather than an agile model of quickly, build, test, release etc.
Eventually they just hired a dev to maintain and make increnetal improvements. the platform is still running.
Personally I'd just take an agile approach. integrate laravel in to the platform to initially just handle auth and hand over the rest of the pages to the old system. laravel's ORM can map to any clunky database, then migrate each of the old pages or features to laravel, one at a time, releasing and testing often until everything has been migrated to laravel and the old site has been phased out.