We’re having an arcade day after the ceremony

Unfortunately, there is no proper way do this, at least not as your wedding reception/ celebration. If you invite people somewhere then you need to pay for them.

A reception is a way for the wedding couple to host their guests to thank them for attending and supporting the wedding. On that note, it's expected that the couple pay for the reception food and drinks. There's no polite way to ask people to come celebrate on their own dime, at least when it comes to etiquette.

The guidance then is that you should have a reception where you host with food/snacks/refreshments and provide a chance for the bride and groom to greet their guests. There should be a separate area where your guests gather and generally no other people wandering in. If an arcade is nearby, it probably should not initially distract from your gathering but could be available for people who stick around. It helps to find the boundaries between entertaining your guests and letting them find something else entertaining.

The answer to your questions is - you don't. You don't invite guests to an event they have to pay for. If people who aren't invited want to later take you out to celebrate you they will offer to do so. Assuming people will want to pay for an arcade day after not being invited to the wedding is likely incorrect.