Inventory Management Systems w/ Website Sales

Hi everyone,
We have an ecommerce website with a fully custom built CMS that manages all of our website data. Invoicing and Point of Sales can get messy with QuickBooks however because it’s an entirely different program which does not reflect the latest website data so things can get out of sync pretty easily. Our accountants also use QuickBooks to handle bookkeeping and help to keep things up to date on the business side of things. So what we would like to do is build an Inventory Management System which can handle all of the sales, invoicing and accounting needs directly from our website. This way all of our data is centralized on our website rather than having broken up between our website and QuickBooks.

Are there any good suggestions for how to do this? We built our website from the ground up using our own CMS for our own needs but maybe there are some pre-built IMS solutions out there? Running on the foundation of our existing CMS is a requirement to whichever solution we go with, and we are definitely not looking at re-building our entire website from the ground up on a different platform - unfortunately that’s just not an option for us. But perhaps these solutions offer a developer API that we can work with? We have checked out LightSpeed Retail but they told us we need to build our website from the ground up on their platform which we are not able to do for our needs.

Retail websites must run into this very same issue. And they must also have a very smart solution which helps to automate a lot of the work and keep things streamlined on their end too.
Any help is greatly appreciated.