Possible solution

This function seems to be working well. I have edited it to incorporate input.

multi.mixedorder <-\(data, cols, na.last = TRUE, decreasing = FALSE) {
  do.call(order, c(
    lapply(cols, function(col) {
      if (is.character(data[[col]])) {
        factor(data[[col]], levels = mixedsort(unique(data[[col]])))
      } else {
        data[[col]]
      }
    }),
    list(na.last = na.last, decreasing = decreasing)
  ))
}

DT[multi.mixedorder(DT, input),]