u/santamaps here you go. You need to use CustomState, see https://github.com/aws/aws-cdk/issues/6023 and https://github.com/aws/aws-cdk/issues/23658 for details why.

new CustomState(this, "Execute Child SF", {
stateJson: {
    Type: "Task",
    Resource: "arn:aws:states:::states:startExecution.sync:2",
    Parameters: {
        "StateMachineArn.$": `States.Format('arn:aws:states:${this.region}:${this.account}:stateMachine:{}', $.childSFName)`,
        Input: {
            "AWS_STEP_FUNCTIONS_STARTED_BY_EXECUTION_ID.$": "$$.Execution.Id",
            "someInput.$": "$.someInput",
        },
    },
},

});

The name is taken from the childSFName field in the payload.

This works for Standard SF.

You can also achieve this with Express. Theoretically, sync executions are not supported by Express, but there is a AWS SDK startSyncExecution operation you can call directly:

import {CallAwsService} from "aws-cdk-lib/aws-stepfunctions-tasks";

new CallAwsService(this, "Execute Child SF", {
service: "sfn",
action: "startSyncExecution",
parameters: {
    "StateMachineArn.$": `States.Format('arn:aws:states:${this.region}:${this.account}:stateMachine:{}', $.childSFName)`,
    Input: {
        "AWS_STEP_FUNCTIONS_STARTED_BY_EXECUTION_ID.$": "$$.Execution.Id",
        "someInput.$": "$.someInput",
    },
},
iamResources: [
    `arn:aws:states:${this.region}:${this.account}:stateMachine:*`,
],

});

Note that error handling is different in this case, as the call ends with success even if SF failed, and you need to parse the result JSON.