Problem:

I'd like to have a way to change the Loop style from the front end. Many websites I've visited have something like this to change the style from grid to list to table etc...

Solution:

There isn't such kind of built-in feature within Toolset Views/Blocks plugin, you might consider to create two views, but display them in different style(grid and list), then use custom JS codes to toggle to different view, for example:

Relevant Documentation:

https://api.jquery.com/toggle/

Problem:
The issue here is that the user is getting the error "Toolset plugin cannot connect to toolset"

Solution:
This usually occurs when there is an issue with the registration key.
As precaution could you attempt to re-register the site by following the instructions in the link below.
https://toolset.com/faq/how-to-install-and-register-toolset/#automatic-installation-once-you-have-types-plugin-installed

Please follow the instructions exactly as this will allow you to automatically get the correct key for the site.