Is there an R function to find the highest value in a row that does not match other names in a column?

Here is the R code I am currently using:

library('tidyverse')
library ('dplyr')

Brian <- c(92.835, 89.035, 99.222, 93.581)
Buckley <- c(75.265, 86.258, 93.972, 96.872)
Chris <- c(91.442, 103.999, 91.291, 92.505)
Catherine <- c(81.244, 73.040, 78.455, 98.972)
David <- c(87.153, 60.062, 62.248, 87.852)
Donald <- c(93.395, 91.905, 102.502, 107.63)
Greg <- c(79.571, 73.702, 67.326, 89.493)
Matt <- c(78.585, 48.074, 81.387, 76.074)
Michael <- c(96.933, 78.709, 82.623, 66.325)

df <- data.frame(Brian, Buckley, Chris, Catherine, David, Donald, Greg, Matt, Michael)
group1 <- data.frame(Brian, Matt, Michael)
group2 <- data.frame(Buckley, Chris, Catherine)
group3 <- data.frame(David, Donald, Greg)

group1a <- group1 %>%
  mutate(Group1 = names(.)[max.col(.)])

group2a <- group2 %>%
  mutate(Group2 = names(.)[max.col(.)])

group3a <- group3 %>%
  mutate(Group3 = names(.)[max.col(.)])

GROUP1 <- dplyr::pull(group1a, 'Group1')
GROUP2 <- dplyr::pull(group2a, 'Group2')
GROUP3 <- dplyr::pull(group3a, 'Group3')

ALL <- cbind(df, GROUP1, GROUP2, GROUP3)

The code shows 4 rows of a much longer table. I've been stumped trying to find code that will allow me to append a column to this table that includes the name of the column from the highest value in the row that is not one of the names in the columns GROUP1, GROUP2, GROUP3. The output column would be headed "GROUP4", and would include Brian, Buckley, Chris, Buckley for these four rows.

I've tried looking through dplyr for codes that would fit this problem, but I'm new to this and have been stumped for a bit.

(possible_names <- select(
  ALL,
  where(is.numeric)
) |> names())
(excl_names <- select(
  ALL,
  where(\(x)!is.numeric(x))
) |> names())

ALL2 <- mutate(rowwise(ALL),
  cols_to_check = list(setdiff(
    possible_names,
    c_across(all_of(excl_names))
  ))
)

(ALL3 <- ALL2 |> mutate(GROUP4 = (\(x){
  x[max.col(pick(x))]})(c_across(cols_to_check))
  ) |> ungroup() |> select(-cols_to_check))


</details>