If you are seeing this error:

This operation is not allowed for inventory items.
Inventory-based listing management is not currently supported by this tool. Please refer to the tool used to create this listing

This is because listings have been created using a different tool e.g. Shopify eBay channel that uses a different set of APIs to what Codisto uses. (Codisto uses the 'Trading' APIs.  APIs are what we use to communicate messages with eBay).

What's the problem?

eBay unfortunately didn't make the new APIs compatible with the trading APIs.  This means that no tool that uses the trading APIs including using eBay directly ('Sell Your Item' flow) can update listings created with the new APIs.

Why doesn't Codisto use the new APIs?

Because the new APIs are not feature complete - you can't do all the operations that merchants need to manage listings using the new APIs.  If we moved to the new APIs there would be many complaints from merchants about things they used to be able to do using Codisto but then couldn't e.g. list using Best Offer.

Why did eBay create a tool using new APIs that aren't feature complete or make them backward compatible with the trading APIs?

Great question and one for eBay management.  If they had not done either of those things there would be no problem but as it is Codisto is stuck with not being able to update listings created with the new APIs.  NOTE this problem extends to editing listings on eBay directly via the normal 'Sell Your Item' flow and all other 3rd party tools!

So what's the solution?

There is no way to update the existing listings using Codisto (or anything except the original tool that used the new APIs).  The only solution is therefore to end the original listings created with the new APIs and create new listings using Codisto.  Ending existing listings can either be done in the old tool (assuming it's still installed) or directly on eBay.  Thankfully creating new listings using Codisto is very quick and easy and then all will remain in sync.  See this article How to list on eBay