Sadly, nano doesn't seem to have any way of bulk-deleting without clobbering the clipboard.
The safest thing to do while staying within the document is probably to paste your clipboard before deleting, then re-cut it again afterwards.
If you have a block of text already selected, then Ctrl+U will paste the clipboard text, including it in your selected block. You can then unmark the pasted text and just cut your originally selected block.
These steps don't preserve your clipboard, exactly, but at least effectively perform a swap between your selection and the clipboard, allowing you to re-cut the lines you had in there before.