Is there anyway I can “claim” records from iNaturalist as my own?

The short answer is: no.

During the transfer of data from iNaturalist, there's no logged in user, which means the resulting records won't have an owner (they will be shown with a crossed-out user icon in the Explore page). Thus, using the same name in your iNaturalist account won't make any difference, because the recorder name won't be linked to a specific iRecord account (it will instead display "Input by admin, core"). These records aren't editable in iRecord, but while they're awaiting review, any changes made in iNaturalist should continue to flow over to iRecord. However, once the review is done, all updates from iNaturalist will end. In addition, there's no mechanism for sending feedback to iNaturalist, and since the imported records don't belong to a particular iRecord account, they won't generate any notifications either.

I suppose one work-around to all this would be to create a search filter for the relevant records. So something like: Source - iNaturalist; Who - [your name]; When - date of last edit or verification / max. age 1 month. You can then set yourself a reminder to review this once a month (or whatever).

PS: one annoyance I discovered whilst experimenting with the filters is that there seems to be no option to search for names containing spaces. Rather stupdily, it insists on searching for [first name] OR [last name], which will obviously get far more hits than you want (unless one of your names happens to be unique). Fortunately, the recorder search-term can be manually edited using the text boxes at the top of the results list, and that does filter on whole names.

Thank you!