How can I get a SVN diff from the point a branch was started to HEAD?

Roughly

svn log --diff --stop-on-copy https://subversion/.../branches/BR-2-7-3

You are correct, it is a matter of doing a diff between the revision number of when the branch was created and the revision number of the version you wish to compare it to (in this case it is probably the latest commit on the branch)

You will need to perform this in two steps:

1) Find the revision number of when the branch was created and the revision number of the latest commit

svn log --stop-on-copy <branch location eg. "https://subversion/.../branches/BR-2-7-3">

This will show you a log of all the commits on the branch with the latest one at the top (in the form rYYYY) and the start of the branch at the bottom (in the form rXXXX). Remember the revision number of both of those commits (XXXX and YYYY)

2) Do a diff between those two revisions

svn diff -rXXXX:YYYY <branch location eg. "https://subversion/.../branches/BR-2-7-3">