Forking might be easier to debug.. but I think I might have missed some of the functionality in your script.. something like (totally untested):

   (while true
        do
            inotifywait \
                --exclude '.*\.sw.' \
                --recursive  \
                --event close_write  \
                --event delete \
                --event moved_to \
                --event moved_from \
                   dir1/
             git push
             sleep 0.1
        done
    ) 2>&1 >/var/log/push_log.log &
   
   (while true
        do
          cat << 'EOF' | ssh remotebox
            inotifywait \
                --exclude '.*\.sw.' \
                --recursive  \
                --event close_write  \
                --event delete \
                --event moved_to \
                --event moved_from \
                   dir1/
             git push #??
             sleep 0.1

EOF

        # (second EOF cannot be indented)
        done
    ) 2>&1 >/var/log/pull_log.log &

    echo "Kicked off pull and push processes."
    wait