How do I disable a core installation profile? There is the Allow install profiles to be uninstalled issue that doesn't allow installation profiles to be uninstalled. Is there any way to do this in D7?
(With a little more effort if necessary.)

When you install D7, you typically have to choose an install profile; this could be the "standard" profile shipped with Drupal core. Once installed, the install profile will sit in the "system" table forever, with no obvious way to disable it.

Is there some more hackish way to disable the install profile.

To harmonize the question with the accepted answer even more, I extend the question to: If the install profile cannot or should not be disabled, then can it be replaced at least?
In Drupal 7, a install profile is like a module, but a special one, with two major exceptions:


Any module required by a install profile can be disabled --as you have found-- except the ones required by drupal core (field, filter, node, system, user). See module_disable() function for details, where the dependency on the install profile is explicitly discarded.
The profile used to install a site is also a required module. This is hardcoded in drupal_required_modules().


So a profile can't be uninstalled, because it is a required module.

You can set status to 0 in system table, but you shouldn't, since:


Drupal expects the install profile to be enabled, and rely on that for some operations, none if them critical afaik.
it is a module, it may implement any hook (in its .profile file), or declare any function required by a profile's submodule. Disabling it the hard way may produce misfunction or a php fatal error in those scenarios.


Btw, the install profile used to install the site is stored in the install_profile variable

# drush vget install_profile
install_profile: 'minimal'


You can, if you find it useful for some reason, change the install profile by setting install_profile to any other profile name and enabling it in the system table.