To be honest this question might be a bit outside the scope of this forum. You can use whatever you like to build the server side stuff - it's all the same to Construct (just grab things off the server with the AJAX plugin).
I can't really speak to the merits of Laravel. Personally I would stick to vanilla PHP/MySQL unless you have a good reason to use something like that (most of the server end stuff you'll want to do will likely be quite simple eg. just pulling things out of a database). Most or all of that type of stuff is quite easy to do with straight PHP without needing to use any additional frameworks.
But perhaps somebody who knows more about Laravel can weigh in on this :)