I am suspicious that you will need to use the RDCOMClient assuming your R session is also in Windows. I know of no solution that works from *nix.
Here's an example from this StackOverflow question which uses RDCOMClient to call a macro inside Excel. You could create a macro that refreshes your pivot then use this R code to call that macro:
library(RDCOMClient)
# Open a specific workbook in Excel:
xlApp <- COMCreate("Excel.Application")
xlWbk <- xlApp$Workbooks()$Open("C:\\Temp\\macro_template.xlsm")
# this line of code might be necessary if you want to see your spreadsheet:
xlApp[['Visible']] <- TRUE
# Run the macro called "MyMacro":
xlApp$Run("MyMacro")
# Close the workbook and quit the app:
xlWbk$Close(FALSE)
xlApp$Quit()
This example does not have a save step at the end which you may need to add.