Unless I'm missing something, what you need to do is, for your local shell, to have the system use the php5.6 executable rather than the php 5.4 one?
You need to find (or install) the php you need: for example, say the 5.4 version is in /usr/bin/php; the later one might be in /usr/local/bin/php
If you can't do that, you may be able to install your own version of php in your home directory (i.e. under /home/joeby/php/... ) though that is unfortunately not as simple as it might be.
Once you've located the correct php executable, do one of two things:
1) [This is the preferred solution:]
Modify your shell PATH: e.g. in .bashrc a line like:
PATH=/usr/local/bin:${PATH}
where the /usr/local/bin is the folder you found the correct php in. It matters that this precedes the ${PATH}. Because the system searches for the php binary in the path, and you have only changed the path for your own shell (not for the system), it should mean you run the php you want, but the system sees the version it wants.
2) modify the drush php file. To do this you'd need to install drush as separate files, rather than a .phar file, and then find the file 'drush' and replace the first line. Normally it should look like this:
#!/usr/bin/env php
you can change it to:
#!/usr/local/bin/php
that is, the complete pathname of the executable you want to use. Once this is done remove write permission for yourself, so it is non-trivial for an update to replace the modified file:
chmod a-w /usr/local/bin/php
drush up drupal
works great for me