Stop Safari from opening FaceTime when a site contains a phone number

There is no easy way to circumvent this behavior in Mobile Safari.

I know of no alternative browser that accomplishes what you ask, but perhaps if someone else does they will answer or edit my answer.

You can avoid those sites, or give more space when asked. It's one of those "we can't adjust the wind, but we can adjust the sails or steer the boat elsewhere" kind of things at present.

HTTP Header: Content-Disposition

Have the server include an additional Content-Disposition header when serving the PDFs:

Content-Disposition: attachment; filename="<MyPDF.pdf>"

This provides the browser with a strong suggestion for the desired filename and suffix. See How to encode the filename parameter of Content-Disposition header in HTTP? for a discussion about limitations and implementing this feature.

MIME Type

Additionally, make sure the server's PDF mime type in the Content-Type: header is set correctly. There are variations in use. application/pdf is the recommended type.

The standard MIME type is application/pdf. The assignment is defined in RFC 3778, The application/pdf Media Type, referenced from the MIME Media Types registry.

Test with curl

Use curl -I http://example.com/mypdf.pdf to test and ensure the correct headers are returned by your server. curl can provide credentials to work as a logged in user.

Potentially related question: When sending headers to download a PDF, Safari appends .html