Jake with Edgemesh here. Unfortunately, there is nothing similar as we (along with the other more advanced service workers) are unable to onboard new clients for the Edemesh Client until Shopify completes its updates to the Service Worker scope.
Which really is a shame. We've heard there is a solution in the works but we've no idea on a timeframe (it's been ~11 months thus far).
An alternative is either getting a Cloudflare Enterprise account so you can control the edge, building a headless site ,or Edgemesh Server.
Sorry the Edgemesh client isn't available for new installs on Shopify today ... its pretty awesome
I believe edgemesh is a precaching - user speed optimizer. If that's the case then there are a few precaching apps that are available in the shopify app store ... Page speed booster, Booster page speed, Ai speed , Jumbo page speed , Super speed etc
You can also try Swish page speed booster. It improves the overall speed score.
Unfortunately without a Service Worker, there's no way to 'cache' aside from the browser's default cache. Most 'pre-cache it just visiting a page (like on hover etc) to try to warm the browser cache. With a service worker, you can actually create and control the cache, which can (using the Cache API) be many many times bigger and more effective.
But as I mentioned, Shopify doesn't currently allow Service Workers on the root scope (which you need for caching) for new sites. I'd ask your account partner to see if they have an updated timeline.