I have a carto web map that I would like to be able to toggle between the Carto basemap and satellite imagery. Similar to how this is easily done in ArcGIS Online with
"Basemap Gallery" widget.
That's not possible with a BUILDER embed map, you need to develop a custom CARTO application using CARTO.js or CARTO VL and it's actually independant of any CARTO especific technology since you would be using pure leaflet or mapbox gl code. You can find an example for mapbox here and for leaflet there's another here.