Unfortunately, I think the “TinyMCE Advanced” plugin is not compatible with BuddyPress, which is why it doesn’t work with the multiline text box in BuddyPress profiles.
If you only want to hide the “link” button, the easy way would be doing that with a bit of CSS code if you are familiar with CSS. Adding a “display: none” instruction on that button should do the trick.
If you want all the options that TinyMCE Advanced provides, you can always try to contact the plugin support and ask them if it is possible to make it compatible with BuddyPress multiline text fields:
Hope this helps.