Yes, if you want to exclude participants who report the outcome before the exposure and incorporate the uncertainty in whether a participant with missing data should be included or not, you should do the imputation first and then exclude participants.
The way to do this in MICE would be to use the complete(..., action = "long", include = TRUE) function after creating the imputations with mice(), perform the desired exclusion calculations on each imputed dataset, then recombine the data using as.mids(). Then you can fit your models on the imputed data and pool the results like normal.
EDIT: On second thought, you probably can't do it like this. I think as.mids will throw an error if the number of rows isn't equal in each imputed dataset.
Instead, you can probably put the exclusion code in the with() block. For example, here I remove anyone with jobperf less than 11 before fitting a glm model in the employee dataset.
library(mice)
imps <- mice(employee,printFlag = FALSE)
fits1 <- with(imps,
glm(jobperf ~ IQ + wbeing))
fits2 <- with(imps,
{dat <- data.frame(IQ,wbeing,jobperf)
dat <- dat[dat$jobperf > 10,]
glm(jobperf ~ IQ + wbeing,
data = dat)})
fits1$analyses[[1]]$y
#> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
#> 10 10 10 10 10 10 10 15 11 7 7 10 11 15 10 10 12 14 16 12
fits2$analyses[[1]]$y
#> 8 9 13 14 17 18 19 20
#> 15 11 11 15 12 14 16 12
As you can see, fits2 excludes anyone with a response vector less than 11. This responds to the imputed values, so people with missing values will sometimes be included and sometimes not. Depending on the details of the model fitting function you're using there might be some complications getting it to work, but this should do what you need.
