Unfortunately, no. Soapbox Mailer specifically looks at the "Soapbox Mailer" Campaign record type in order to ensure the proper Campaign and Campaign Members are included in its functionality. You can add your own custom fields to the page layout associated with the Soapbox Mailer Campaign record type, but unfortunately, you can only use the Soapbox Mailer Campaign and Campaign Member record types to send emails via Soapbox Mailer.
For those looking to add new Campaign or Campaign Member custom fields onto the Soapbox Mailer page layouts, a potential solution could be to make a new section on the page layout that clearly states which team of users those fields would be most appropriate.
Happy emailing!