Do you mean Data loader as the manual route?

Also when you meant Custom Setting, if its actual 'custom settings' you can update through Apex easily You may create a Class with Sample method like below and call the Class in Post Refresh settings while you initiate the Refresh.

Sample class:

Public class Postrefreshsample {

Public void myinitmethod()
{
List<CSetting__c> Settings = [select id,Config__c,Value__c from Csetting__c];

  for(CSetting__c CSet: Settings)
  {    
    if(CSet.Config__c == 'SettingA')
         Cset.Value__c = 'myvalueASandbox';
    
    if(CSet.Config__c == 'SettingB')
         Cset.Value__c = 'myvalueBSandbox';
   }
 Update Settings ;
 }
}

by any chance if you meant Custom Metadata type, then you can call them from Apex as well but little more work. below reference might be useful.
https://www.sfdcpanther.com/create-update-custom-metadata-using-apex/

PS:
I deal it little differently though. For every Custom metadata or Custom Settings, we store 2 rows. one for Prod and one For Sandbox.

And in every reference to the Custom Setting in Apex or Formula, we use the Custom Setting record Conditionally.

Eg:

String labeltoquery;

if(UserInfo.getOrganizationId == 'a0L2M00000NZsample') // this would be Prod org id which never changes.
   labeltoquery = 'SettingA_Prod';    

else 
 labeltoquery = 'SettingA_Sandbox';

then while using querying the Custom setting,

CSetting__c Settings = [select id,Config__c,Value__c from Csetting__c where Config__c =: labeltoquery ];

This way you dont have to update the Custom Setting after every refresh.