You mean like this?

library(leaflet)
library(htmlwidgets)

leaflet() %>%
  addTiles() %>%
  setView(lng = -121.2722, lat = 38.1341, zoom = 10) %>%
  addMiniMap(tiles = providers$OpenStreetMap, position = 'topleft', width = 250, height = 420) %>%
  onRender("
function(el, x) {
  var map = this;
  if (map.minimap && map.minimap._miniMap) {
    var mini = map.minimap._miniMap;

    var style = document.createElement('style');
    style.innerHTML = `
      .transparent-tooltip {
        background-color: transparent !important;
        border: none !important;
        box-shadow: none !important;
        font-weight: bold;
      }
    `;
    document.head.appendChild(style);

    L.circleMarker([37.79577, -121.58426], { radius: 2 })
      .setStyle({ color: 'green' })
      .bindTooltip('mylabel', {
        permanent: true,
        direction: 'right',
        className: 'transparent-tooltip'
      })
      .addTo(mini);
  }
}
")