Need to add Shipping method to order summary

I found a piece of code to add it but where do I add it? It seems like it's only available in the Business version because I don't see any front-end email files except forms in the free one

Yes the edition of the emails is only available in the business version of HikaShop

If I do decide to upgrade to the business version, are there any issues to be aware of? We have a live shop running and I'm concerned that if I upgrade, then it'll overwrite any configurations thats already set

If you did you customizations through our views editing system you shouldn't loose anything.
You can still make a backup of your website before upgrading but the Business version should just bring you more features