How to implement Fragment Caching (W3TC) in Wordpress
Based on the description of your problem this isn't being caused by W3TC in general, but rather by the Page Cache functionality that W3TC provides. In the W3TC settings within WordPress, shut off page cache and only use the fragment cache and you will not have this problem.
Or, load in this special content with javascript after the initial page load on mobile devices so that the page can be cached with W3TC without causing problems.