Using dplyr instead of lapply

I have a dataframe with a bunch of start and end dates and I am looping through a list of dates and seeing how many rows in my dataframe are 'open' during that date on the list (i.e. the start date has happened but the end date hasn't).

I am curently doing this using lapply but I was wondering if it could be done in dplyr instead and if there is any benefit in terms of memory and speed (actual dataframe is 1.5M rows).

      RollingDateRange <- seq(Sys.Date()-15, Sys.Date(), by="days")
      temp <- data.frame(RollingDateRange)

      dat <- data.frame(
        Order = c(1,1,1,2,2,2,3,3,3), 
        Code = c("Green","Yellow","Blue","Yellow","Yellow","Red","Purple","Green","Blue"),
        Start.Date = as.Date(c("2020-02-01","2020-02-02","2020-02-03","2020-02-01","2020-02-02","2020-02-03","2020-02-01","2020-02-02","2020-02-03")),
        End.Date = as.Date(c("2020-02-02","2020-02-08",NA,"2020-02-07","2020-02-06",NA,"2020-02-03","2020-02-08","2020-02-06")),
        Count = c(1,1,1,1,1,1,1,1,1),
        stringsAsFactors = FALSE)

      temp$Count <- lapply(temp$RollingDateRange, function(d){
        b <- dat[((dat$Start.Date <= d) & (dat$End.Date >= d)) | ((dat$Start.Date <= d) & (is.na(dat$End.Date))),]

        total <- sum(b$Count, na.rm = TRUE)
      })

Output:

> temp
   RollingDateRange Count
1        2020-01-25     0
2        2020-01-26     0
3        2020-01-27     0
4        2020-01-28     0
5        2020-01-29     0
6        2020-01-30     0
7        2020-01-31     0
8        2020-02-01     3
9        2020-02-02     6
10       2020-02-03     8
11       2020-02-04     7
12       2020-02-05     7
13       2020-02-06     7
14       2020-02-07     5
15       2020-02-08     4
16       2020-02-09     2