I noticed that your screenshot about The inspect section of the modal view .
The element has an attribute original_url which is the original url of the pdf file.
And reference to this answer, you can try to directly download the pdf file by configurating webdriver.ChromeOptions().
So in your case it can be like this:
from selenium import webdriver
profile = {
'download.prompt_for_download': False,
'download.default_directory': '/path/to/download/the/pdf',
'download.directory_upgrade': True,
'plugins.always_open_pdf_externally': True,
}
options = webdriver.ChromeOptions()
options.add_experimental_option('prefs', profile)
driver = webdriver.Chrome(options=options)
driver.get('your_url')
# Your code to handle the Captcha
# When you open the modal dialog box
pdf_url = driver.find_element("id", "plugin").get_attribute("original_url")
driver.get(pdf_url)
# Chrome will download the PDF automatically