Satin finishes nowadays are obtained by adding 'wrinkling' chemicals to
the gloss mix.
Long before these chemicals were available, one obtained a satin finish
by hand rubbing gloss varnish with a mixture of pumice and/or
'rotten-stone'. This hand-rubbing process was the used for hundreds of
years before the advent of chemical addition. This hand rubbing process
can also be used to 'satinize' your existing gloss repair. All you
will need is several grades of pumice, rottenstone, a soft oil soaked
rag. Rotten stone and varying grades of pumice can be obtained in an
old-fashioned paint store. The technique of hand rubbing can be
obtained by doing a websearch for: "rotten stone" + "furniture
refinishing". You'll probably have to 'flat sand' the repaired areas
with 1000 grit wet&dry sandpaper first and then hand rub the entire
panel to get the effect correct. Hand rubbing is fairly easy and not
too labor intensive.