Unable to override taskRoleArn when running ECS task from Lambda

I have a Lambda function that is supposed to pass its own permissions to the code running in an ECS task. It looks like this:

    ecs_parameters = {
        "cluster": ...,
        "launchType": "FARGATE",
        "networkConfiguration": ...,
        "overrides": {
            "taskRoleArn": boto3.client("sts").get_caller_identity().get("Arn"),
           ...
        },
        "platformVersion": "LATEST",
        "taskDefinition": f"my-task-definition-{STAGE}",
    }
    response = ecs.run_task(**ecs_parameters)

When I run this in Lambda, i get this error:

"errorMessage": "An error occurred (ClientException) when calling the RunTask operation: ECS was unable to assume the role 'arn:aws:sts::787364832896:assumed-role/my-lambda-role...' that was provided for this task. Please verify that the role being passed has the proper trust relationship and permissions and that your IAM user has permissions to pass this role."

If I change the task definition in ECS to use my-lambda-role as the task role, it works. It's specifically when I try to override the task role from Lambda that it breaks.

The Lambda role has the AWSLambdaBasicExecutionRole policy and also an inline policy that grants it ecs:runTask and iam:PassRole. It has a trust relationship that looks like:

"Effect": "Allow",
"Principal": {
  "Service": [
      "ecs.amazonaws.com",
      "lambda.amazonaws.com",
      "ecs-tasks.amazonaws.com"
  ]
},
"Action": "sts:AssumeRole"

The task definition has a policy that grants it sts:AssumeRole and iam:PassRole, and a trust relationship that looks like:

"Effect": "Allow",
"Principal": {
   "Service": "ecs-tasks.amazonaws.com",
    "AWS": "arn:aws:iam::account-ID:role/aws-service-role/ecs.amazonaws.com/AWSServiceRoleForECS"
},
"Action": "sts:AssumeRole"

How do I allow the Lambda function to pass the role to ECS, and ECS to assume the role it's been given?

P.S. - I know a lot of these permissions are overkill, so let me know if there are any I can get rid of :) Thanks!