I only have experience with Fortrabbit free development plan, but it is easily up gradable if needed, than also Pagoda box, which allows you to choose your server configuration, ie bandwidth, storage, databases... And also Laravel Forge which is "Tuned For Laravel." as it says on their website.

Simple deployment, customizable...

Have you considered Amazon? I'm hoping to go with AWS reserved EC2 instance (via Elastic Beanstalk). You pay for what you use...but we'll have to see what the monthly charges end up at for the app.

Have you tried Cloudways before? They are offering managed hosting for Laravel and other PHP based applications. You can use their platform to deploy application on Amazon Web Services EC2, Google Compute Engine, DigitalOcean, and Vultr in just a single click. Their platform supports Varnish, Memcached, Apache and Nginx stack.

I have one setup on godaddy < 4.00 per month. But if dealing with sensitive information, you may need to look for a dedicated hosting server. More expensive though. Byethost is another, they even offer a free plan.

HOSTING LARAVEL It offers web hosting for your projects in Laravel Framework all versions (4.x up to 5.4), we have support for PHP versions 5.3, 5.4, 5.5, 5.6, 7.0 and 7.1, up to 20 times faster thanks to the storage technology SSD. Http://hostinglaravel.com/