I did some testing and was able to get the delete function to work with the rest API.
The first call below creates a test key on the robot.cfg for the controller.
POST /rest/probe/PrimaryUIM_domain/PrimaryUIM_hub/PrimaryUIM/controller/callback/probe_config_set HTTP/1.1
Content-Type: application/xml
Authorization: Basic YWRtaW5pc3RyYXRvcjoxMjM0NTY=
Cache-Control: no-cache
<callbackrequest>
<timeout>50000</timeout>
<parameters>
<name>name</name>
<type>string</type>
<value>controller</value>
</parameters>
<parameters>
<name>section</name>
<type>string</type>
<value>test</value>
</parameters>
<parameters>
<name>key</name>
<type>string</type>
<value>Testkey</value>
</parameters>
<parameters>
<name>value</name>
<type>string</type>
<value>MytestKeyValue</value>
</parameters>
</callbackrequest>
this created the following in my robot.cfg
<controller>
robotname = PrimaryUIM
hubip = ***.***.***.***
robotip = ***.***.***.***
first_probe_port = 48000
set_qos_source = yes
ip_version = ipv4 ipv6
domain = PrimaryUIM_domain
hub = PrimaryUIM_hub
hubrobotname = PrimaryUIM
hubport = 48002
access_0 = 0
access_1 = 1
access_2 = 2
access_3 = 3
access_4 = 4
<environment>
NIM_JRE_HOME = jre/jre8u102
NIM_JRE_HOME_1_8 = jre/jre8u102
</environment>
</controller>
<test>
Testkey = MytestKeyValue
</test>
The second call remove the section all together:
POST /rest/probe/PrimaryUIM_domain/PrimaryUIM_hub/PrimaryUIM/controller/callback/probe_config_set HTTP/1.1
Content-Type: application/xml
Authorization: Basic YWRtaW5pc3RyYXRvcjoxMjM0NTY=
Cache-Control: no-cache
<callbackrequest>
<timeout>50000</timeout>
<parameters>
<name>name</name>
<type>string</type>
<value>controller</value>
</parameters>
<parameters>
<name>section</name>
<type>string</type>
<value>test</value>
</parameters>
</callbackrequest>
Resulting robot.cfg
<controller>
robotname = PrimaryUIM
hubip = ***.***.***.***
robotip = ***.***.***.***
first_probe_port = 48000
set_qos_source = yes
ip_version = ipv4 ipv6
domain = PrimaryUIM_domain
hub = PrimaryUIM_hub
hubrobotname = PrimaryUIM
hubport = 48002
access_0 = 0
access_1 = 1
access_2 = 2
access_3 = 3
access_4 = 4
<environment>
NIM_JRE_HOME = jre/jre8u102
NIM_JRE_HOME_1_8 = jre/jre8u102
</environment>
</controller>
Notes do create a section under another secion you would change
<parameters>
<name>section</name>
<type>string</type>
<value>test</value>
</parameters>
to
<parameters>
<name>section</name>
<type>string</type>
<value>environment/test</value>
</parameters>
Hope this helps