I'm due for a flight review shortly, so I figure I'll start a new logbook
with that entry.

I'm wondering what anyone else does regarding endorsements. I have a high
performance endorsement, and I don't know how I should carry it forward.

Should I photocopy the page from the old logbook and paste/tape it in the
back of the new book?

I wouldn't change logbooks just because of a flight review (or even a
new rating). Just keep using the old one until you run out of pages.
But in any case, when you go to a new logbook, make a note in the new
book of the dates and CFIs for all endorsements still needed, and of
which logbook they are actually located in. You don't generally have to
carry your endorsements with you, just have them. (You don't have to
carry your logbook either, in general).

There is no harm in pasting a photocopy of an old endorsement in the new
logbook.

There is great benefit in photocopying (or scanning) your entire logbook
and putting the copy in a safe place. Logbooks do get lost. Mine was
stolen some twenty years ago. I was fortunate in that I kept a
photocopy of every page.

Jose
--
Get high on gasoline: fly an airplane.
for Email, make the obvious change in the address.