It looks like they automatically switch to full-page view in small screens. What you can do is override that behavior with the same styles they are using for side-modal. Add this to your global css:
.snipcart-modal__container {
width: 35% !important;
max-width: 540px !important;
margin-left: auto !important;
right: 0 !important;
}
.snipcart-modal__container.snipcart-layout--large {
width: 100% !important;
max-width: 100% !important;
}
You can also remove data-config-modal-style="side" from HTML now, since you don't need it anymore.