$fullAccess should cause the conditions to be ignored, but won't work for nesting behaviors. Example as follows.
Code:
$avatar = $cbUser->getField( 'avatar', null, 'html', 'none', 'list', 0, true );
Worst case you can try setting an internal flag as shown below to force the bypassing.
Code:
$avatar = $cbUser->getField( 'avatar', null, 'html', 'none', 'list', 0, true, [ '_noCondition' => true ] );
_noCondition is a specific behavior in CB Conditional for flagging conditions off. You can also just access the data directly through $user and bypass everything to get the database stored value. Example as follows.
Code:
$name = $user->getString( 'name', '' );