It seems you are trying to deploy from you home directory. In this case Your bundle may contain too many files if you don't select them.
First, Good practice is to use rstudio project. This will be easier to deploy from the IDE. Indeed there is publish button that you may already use. This integration of rsconnect with the IDE help you deploy. The menu changes depenging from which window you click : source or viewer.
This menu should help you select which files to put in your bundle . Just put the files that you really need.
Note you can also use rsconnect
directly. There are argument for files selection in the deploy functions.
So to prevent from sending more than 1000 files, look at you bundle, adjust the files selection to put only the necessary, and try to republish.