I don't think you can do what you're suggesting. Maybe with some VBA programming, but that's out of my ability. The filter just displays the cells that meet a given criteria. All the other non-visible rows are still there, just hidden. So, any copy, paste, etc. affects the hidden rows as well.

You could add a column to your spreadsheet that checks each row to see if there is a match for your filtered criteria and then the formula would display "Match" or "1" or something that shows it's a match. Then you could sort on the new column and that would collect all of your desired records together so you could copy/paste etc. without having hidden, intervening records. Kind of a clumsy workaround, but maybe better than nothing.