addMiniMap()
takes the tiles
parameter which allows you to specify a URL for map tiles or use one of the pre-defined map tile providers, so just adjust your code to the following:
leaflet() |>
addProviderTiles("CartoDB.Positron") %>%
setView(lng = 13.4, lat = 52.52, zoom = 14) %>%
addMiniMap(width = 150, height = 180, position = "topleft", tiles = "Esri.NatGeoWorldMap")