The best replacement for best wishes is congratulations.

The best alternatives to best wishes are rated as following:
★★★★★  congratulations
★★★★  sincerely
★★★  best regards
★★  good luck
★★  congratulations engagement
★  congratulations for the bride
★  best
★  all the best
★  congratulations etiquette
★  best wishes
★  congratulations wedding etiquette
  congrats
  best of wishes
  congratulations bride
  congratulations to the bride
  congratulations to bride
  congratulations wedding
  congratulations on wedding

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Speaking about best wishes, one probably shouldn't forget about regards.

Other search terms related to best wishes and their ratings are:
★★★★★ regards
★★ happy birthday
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best wishes
congratulations
sincerely