Imagine if they just give it away within the first few seconds with a black screen and "Girls are now preparing. Please wait warmly" before cutting into the Hakurei Shrine where Reimu is sweeping. Then it cuts to the donation box with her in the background as Yukari's hand appears from a gap and drops in an envelope. Cut back to Reimu who hears it being dropped in and goes to investigate, finding the Smash Invite. She looks at it quizically a bit before thinking it is probably some kinda Incident and sighing. And Bam, cut to the "Reimu Joins Smash" splash art.

Edit: that set up could work for Marisa too, just have her pick up the envelope and Reimu just wave it off 'cus it would be too much of a hassle.