As far as I know, you should have the browser installed, as I believe the driver is just supposed to be used to control the browser... though your success with it working without it makes me doubt that theory.

As for the missing class attribute issue... hmm, it's possible that the class isn't there OR it's not parsing the .html correctly. Have you used driver.page_source to verify that attribute exists in the html when selenium gets it? You can do that by waiting for it to fetch the third page, then do print(driver.page_source). Warning: it will be a LOT of output, so you might just want to save that output to a file, then do

cat file.txt | grep FS03250425_BCN03A20

to see if it's where you expect it.