Password Protected Website to PDF

I would recommend that you don't use the method that you're currently using.

(NOTE: In all suggestions below, It is also very important that you become familiarized with the type of font encoding used in the HTML web pages so that you don't get PDF irrepairable errors with embedded text or alternate text after the export finalizes.  In addition, it is worth noting that this process is extremely slow and long. The download performs in bytes at a time. Consider a fast Internet connection over Ethernet. Do not perform this method using a Wi-Fi enabled computer.)

Login to the password protected website but use any of these three methods to download the entire website: 

• Suggestion 1 - Use Save As option directly from web browser and select " Webpage, Complete (.htm, html)"  . Then convert the downloaded HTML file to PDF using Acrobat "Export To PDF"
•  Suggestion 2 - Use the Adobe Acrobat PDF maker extension. For the results that you're looking in terms of splitting chunks of webpages or specific sections use the extension with Internet Explorer or Mozzilla Firefox. According to Adobe Helpx guidance these are the only two web browsers that allow to perform specific selections or sections of a web page. Otherwise, every other nethod just try to fetch and download everything at once.
• Suggestion 3 - Use the same method that you're already familiarized with.  But since you've mentioned about the password login issue, this is how I would recommend to do it:

1. Log-in to the password protected website, and then Use HTTrack or a similar tool to protect the whole integrity of the HTML website and download for offline use. This suggestion is to avoid your Acrobat from crashing while it fetches the large HTML content that you intend to export to PDF.
2. With Acrobat open, go to tools and select "Create PDF". From the Wizard select "Web Page".  It is very important that you select the HTML file that was downloaded for off-line use. Do not  to use the fetching opion by entering a URL. 
3. Tick the box "Capture Multiple Levels" and type in 5. Click on "Advanced Settings" to configure font type and encoding.  When you're done setting up your preferences click on "Create".

See slides below:

Before you click on Create, click on "Advanced Settings" 

This is a slide of showing the long and almost unresposnive process when you fetch a URL. I strongly recommend staying away from this option and download the website for off-line use in a separate process (Avoid using Acrobat for this or you won't be able to resolve your issue).

However, if you still prefer to fetch a large password protected website without the aid of HTML download tools ,  you may want to consider a javascript action or batch sequence which seems doable , in which case, and assuming that the secuirty enforced in that website also  uses javasript, then  you must contact the webmaster of such website and see if what you're trying to do is permissible to begin with.

You may end up bumping into other secured levels of the website, not just the first page.

Consider this additional information: 

https://www.yourhtmlsource.com/sitemanagement/passwordprotection.html

http://www.javascriptkit.com/script/cut10.shtml

1. Go to site URL and log-in with your browser.
2. Go to File > Print > and select your printer: Adobe PDF. 
3. Navigate to the next site page and repeat Step 2.

I'm sorry, I have no automated way to do this for an entire domain.

You can only do it from within the browser itself, not from Acrobat.

In which webbrowser is this happening?

Are we talking about the Adobe Acrobat PDF Maker extension?