Lightdm-gtk-greeter uses a different icon set. This question on launchpad explains the naming theme for the icons. I'm using lubuntu 14.04 and the aforementioned icons are located in
If you already have an icon (svg file) you want to use, move it to that location and name it
sessionname_badge-symbolic.svg where sessionname is the name of your session (the session name is found in
/usr/share/xsessions), budgie in your case. Make sure to update the icon cache afterwards or the greeter still won't use it.
sudo gtk-update-icon-cache /usr/share/icons/hicolor
That should be it.
If that doesn't work, your icons may be in a different location, or there may not be a conf file for the particular session in
/etc/lightdm. See the
/etc/lightdm/lightdm-gtk-greeter-ubuntu.conf file and LightDM - Ubuntu Wiki for more details on writing up a .conf file. Hope this helps.