Gig Performer is notarized by Apple and can be opened in Catalina.
As of version 3.6, Gig Performer itself runs perfectly fine in Catalina. You will most likely be prompted to allow access to the Microphone and you must allow this.
The prompt is a little bit misleading, as it really is asking permission to allow any audio input which includes vocals or guitar coming from your external audio interface, not your Mac’s internal Microphone. For more information, see this article about no audio.
However, be very careful about using plugins that have not yet been updated to work in Catalina. Various plugin companies have issued statements recommending that users do not upgrade to Catalina just yet.