A photocopy (or more) of the logbook, kept at a separate location (locations) is a much better and easier way to reconstruct a logbook. I have one at work, everytime it changes I make a new last page. The copy will show very important detail such as CFI names, signatures, and certificate numbers, endorsements, BFR's, airports, instrument time etc, etc., that you wont get from an invoice.
If you are a professional pilot and want more assurance that the FAA will accept your reconstruction, you can have the copy certified by a notary and keep that copy in a safe deposit box. Or, keep the original in a safe deposit box and fly the copy, only bring out the original to update it or when you will be recieving an endorsement of some kind.
Save the invoices if you are a pack rat, otherwise it's just clutter.