If you specify a list of targets for a hot backup, "file" and "table" are the only supported top-level URIs, you can't do a hot backup using an "lsm:name" URI.
I think this is an argument for removing the top-level "lsm" type entirely, it makes things harder than they should be – I'd rather require the underlying "type=lsm" configuration instead.
Absent that, the backup cursor code needs modification.