I'm not well-versed enough in dplyr evaluation rules to know why your version doesn't work, but it looks like it works if you specify the functions directly in the mutate
library(tidyverse)
df <- tibble(a = 2, b = 4, c = 8)
df %>%
mutate(across(everything(), list(div_a = ~ .x / a,
div_b = ~ .x / b)))
#> # A tibble: 1 x 9
#> a b c a_div_a a_div_b b_div_a b_div_b c_div_a c_div_b
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2 4 8 1 0.5 2 1 4 2