A critical action that messes everything up is changing the Record Type. You could be looking at a new page layout (with mandatory and readonly fields), as well as new picklist dependencies... There's a reason you choose it initially before seeing any other layouts. And why is changing the record type a special button rather than a field displayed during a typical edit action?
If you know what you're doing and are convinced that the requirements will not change, you'll be able to depict the new layouts, picklists, and so on.
getObjectInfo can be used to retrieve the Map of record kinds, among other things.
Build a lightning-combobox with this data that changes a helper variable onchange (ideally, you'd have that variable bound to then submit).