Hello.
Congrats on your project.
My suggestion, (for what it's worth), is that you should host your site on Laravel Forge, or at least on a platform that allows you to utilize all the features of Laravel.
You've taken the time to build your project on the best PHP framework, why host it someplace that doesn't let you manage any of the things you need to?
You'll be extremely limited in what you can do in a shared hosting environment, and while it'll cost you a few dollars more per month to use Forge and either DigitalOcean or Linode, you'll have a much better experience and be able to take advantage of the true power of what you've built.
As someone who always tried to host on the cheap in the past, I can tell you first hand, once I became aware of Forge, my life became a whole lot better.
I host everything there, development projects, live projects, and not just Laravel, I have several Ghost blogs and static sites, API documentation, and so on.
When Taylor builds something, it works, and usually works like a friggin champ. Give it a look before you mess with that horrible shared hosting.
My 2 cents. =)