To change Logo for mobile homepage in Squarespace, you can follow these.
#1. First, you need to upload another logo and find its url. Follow #1 in this guide to find url
In my example, we will have url:
https://images.squarespace-cdn.com/content/v1/6672ecf757a5d945cabc3965/1d45ed6f-3a9a-4523-9320-1633c8d98b90/st-peters-church-3536449_1280.jpg?format=500w
#2. Use this code to Custom CSS box
body.homepage div.header-display-mobile img {
content: url(https://images.squarespace-cdn.com/content/v1/6672ecf757a5d945cabc3965/1d45ed6f-3a9a-4523-9320-1633c8d98b90/st-peters-church-3536449_1280.jpg?format=500w);
}