Your Main KPI - For eCommerce the main KPI is ROAS, and for Lead Gen the main KPI is CPL.
It is likely that LinkedIn campaigns to warm leads (aka remarketing) could be a good fit, but LinkedIn campaigns to cold leads is only a good fit for a few narrow industries.
If you have a large customer list, and if your Google account is eligible for "customer match," then you can implement some cross-selling and retargeting campaigns. Because the audience size is limited and much smaller than an audience found on Google, we won't need a large budget for this strategy. After about a month or two of gathering conversion data, we can also create a "similar audience" based on the customer list. Similar audiences are often effective, and we will incorporate it into our proactive targeting campaigns.