In the current workflow if you make any changes to an item or group of items. The status changes to "Ready for Approval" and removes the item from the web. If a customer was looking to purchase this item and a change was being made this would remove the item and it would not be purchasable until it is published again. The risk would be a potential lost sale if the item was removed.
Your current recommended process is to load products, clean them in PIM - assign categories, then publish. That is great if you are in an environment that the data is accessible and clean when you receive it, or you have time to enhance it before it is published.
I would like to recommend a workaround to be able to publish an item, so it is sellable while data is being enhanced in the background. In other PIM platforms, I have seen the "check out process". Check out an item before you can edit it, leaving a sellable shadow or imprint on the web. When the edit is complete you would save the changes, check the product back in, and publish the changes. When the refresh would run, it would refresh the item. The Check Out Process would be fine, but it is also a little cumbersome. Would it be possible to have an "edit" mode of some type that would leave a sellable imprint on the web, so data enhancements can be made without impacting your offering? This would also need to be available in the import to be able to export out, enhance data, and import changes back in. Not just for optimized or enhanced data, but general maintenance as well.
This capability already exists, and is dependent on the way you setup your data governance. We recommend using 'required' properties on templates to indicate data that is absolutely required before you publish a product to your site. You can set additional properties as 'recommended' on templates to indicate data that is important to clean up but you may not require it to get the product live. You can filter for products that are missing recommended data after they have been published and then make changes that will be picked up the next time the product is published. Any changes in PIM will not take effect in commerce until the commerce job is run, so these products will not become unsellable when edited. Products would only be removed from the site if you make changes like removing them from categories or setting a deactivate date and then publishing these to commerce.
I think you might be running the wrong PIM job if this is a problem.