Here is the initial design idea I came up with after reviewing your brief and neon sign. The first thing I noticed is that you will likely be printing a lot of t-shirts, which is why I limited myself to solid clean colors (gradients, shadows, textures, etc. will not print onto silkscreened shirts.) The current one I have shown here is a 3-color design (red, yellow, white) on blue shirts/background--having it at 3 colors will allow for easy and affordable printing. I also included a distressed/textured layer to give the shirts a cool vintage look and feel; this is on it's own layer and can be easily turned on or off as needed!
Let me know your thoughts, both in terms of content and color and I will be happy to provide revisions as needed. Thanks for looking! :)
Here is a version for white background/shirts, with the same general layout. As you can see, changing the colors around is quick and easy with this design. Also, I forgot to mention: both the burger and hot dog portion of the sign are in the shape of themselves (just like the ice cream!) to maintain that fun playful aesthetic that your neon sign has.