It seems I have found the key to solving this problem: I need to await my database updates and then refresh my dataset. It seems my dataset refresh was getting ahead of itself.
await wixData.update(DATABASE, item);
The repeater updates fine now that the refresh actually waits for the database changes to complete before refreshing; no need to use repeater.data to assign the data to the repeater, I can still use the dataset that I connected!
Thanks for the help, everyone 