On the page you linked to there is a list of caching plugins that it claims will work with the mobile redirection setting:
- WP Rocket
- WP Super Cache
- W3 Total Cache
- Litespeed cache
- WP Fastest Cache
- SG optimizer
WP Rocket is a paid plugin, and Litespeed and SG optimizer are for specialized types of hosting companies, but any of those other plugins would be worth looking into.
Your problem with the caching plugin breaking your website isn’t unique to W3. Any caching plugin is likely to create problems until you fine tune the settings. What you’ll want to do is to start with all of the caching options disabled, and then go through and start activating options until you find the option(s) that are problematic. Do a hard refresh (ctrl + shift + f5) after making any changes to clear out your local cache, and make sure you don’t have any other caching systems such as Varnish or CloudFlare turned on while your testing things out. It’s a pain to do, but it is well worth the performance benefit that you get.