When doing a callout it has to happen before the DML action. If it happens after the DML it will throw an error or will not execute. Workaround I used is to manually keep track of the mock response, and call the respond method to intercept Http.send when running a test. By checking for Test.isRunningTest() && (mock!=null) we can send the mock response.

Checking for test running and returning mock response

	public static HttpCalloutMock mock = null;
	public HttpResponse callApiEndpoint(String apiEndpoint, String method, Object aPayload) {
		HttpRequest req = new HttpRequest();
    try {
      if (apiEndpoint != null) {
        req.setTimeout(120000);
        req.setMethod(method);
        setAuthHeader(req);
        req.setEndpoint(WebCustomSettings.Heroku_API_URL__c + apiEndpoint);
				if (aPayload!=null)
        	req.setBody(String.valueOf(aPayload));
        System.debug('Sending api request to endpoint' + apiEndpoint);
				if (Test.isRunningTest() && (mock!=null)) {
					 return mock.respond(req);
			 	} else {
					Http http = new Http();
	        return http.send(req);
				}

      } else {
        System.debug('Service apiEndpoint and payload must not be null');
        throw new Rest_Exception(ResponseCodes_Mgr.getCode('HEROKU_REQUEST_CALLOUT_FAILED'));
      }
    } catch (Exception ex) {
      System.debug('ERROR sending sync request to Heroku: ' + ex);
      List<String> theArgs = new List<String>();
      theArgs.add('Api');
      theArgs.add(req.getEndpoint());
      throw new Rest_Exception(ResponseCodes_Mgr.getCode('HEROKU_REQUEST_CALLOUT_FAILED', ex, theArgs));
    }
  }

Test class

  Test.startTest();
        System.runAs(new User(Id=userWithAccountContact.get('UserId'))){
          //Setting the mock variable and it's also a test so will return the mock response
          Heroku_Services_Api.mock = new Heroku_Services_Api_Mock(200, 'Complete', '{}',null);
          req.requestBody = Blob.valueOf('{"id":"'+ account.Id +'", "clientId":"' + userWithAccountContact.get('ContactId') + '", "amount":122233,"endDate":"2022-01-01"}');
          req.requestURI = '/v1/account';
          req.httpMethod = 'PATCH';
          RestContext.request = req;
          RestContext.response = res;

          try{
            Rest_Dispatcher.doPatch();
          } catch(Rest_Exception ex){
            System.assertEquals(ex.itsRespCode.formattedSystemMessage, 'blah');
          }

          System.assert(res.responseBody!=null);
          System.assertEquals(res.statusCode, 200);
      }
      Test.stopTest();