How to use with HCP Vault?

I'm following the quickstart and following this video [AWS re:Invent - Using Vault with AWS Lambda and More](https://www.youtube.com/watch?v=U3zHvg-2WJw) with a Vault server hosted on Hashicorp Cloud Platform.

I'm stuck at the third step of the guide, running `vault write auth/aws/role/vault-lambda-role`

Around the minute [16:12](https://youtu.be/U3zHvg-2WJw?t=972) of the video I linked a similar issue is mentioned, but the solution is not very clear.

These are the commands I ran:

Configure my Vault client
```sh
export VAULT_ADDR=""
export VAULT_TOKEN=""
export VAULT_NAMESPACE=admin
```
Enable AWS auth backend and configure it as the guide says
```sh
vault auth enable aws
Success! Enabled aws auth method at: aws/
```
```sh
vault write -force auth/aws/config/client
Success! Data written to: auth/aws/config/client
```
Add a config STS since my lambda and its role are not in the same account as my user:
```sh
vault write auth/aws/config/sts/*** auth_type=iam sts_role="arn:aws:iam::***:role/vault-lambda-role"
Success! Data written to: auth/aws/config/sts/***
```
Then run the final command from the guide
```sh
vault write auth/aws/role/vault-lambda-role \
auth_type=iam \
bound_iam_principal_arn="arn:aws:iam::***:role/vault-lambda-role" \
policies="serverless" \
ttl=1h
Error writing data to auth/aws/role/vault-lambda-role: Error making API request.

URL: PUT https://***.aws.hashicorp.cloud:8200/v1/auth/aws/role/vault-lambda-role
Code: 400. Errors:

* unable to resolve ARN "arn:aws:iam::***:role/vault-lambda-role" to internal ID: AccessDenied: User: arn:aws:sts::285268573241:assumed-role/HCP-Vault-333b91a7-4002-4181-b026-ca105a6eca86-VaultNode/i-06f09e6fc27076955 is not authorized to perform: sts:AssumeRole on resource: arn:aws:iam::***:role/vault-lambda-role
status code: 403, request id: 3044253b-6b33-4f8b-93a5-935c600fd211
```
This is the assume role policy of my role:
```hcl
data "aws_iam_policy_document" "assume_role" {
statement {
actions = ["sts:AssumeRole"]

principals {
type = "Service"
identifiers = ["lambda.amazonaws.com"]
}
}
}
```
I already tried allowing the assumed role from the error message, with no success:
```hcl
data "aws_iam_policy_document" "assume_role" {
statement {
actions = ["sts:AssumeRole"]

principals {
type = "Service"
identifiers = ["lambda.amazonaws.com"]
}
}

statement {
actions = ["sts:AssumeRole"]

principals {
type = "AWS"
identifiers = [
"arn:aws:sts::285268573241:assumed-role/HCP-Vault-333b91a7-4002-4181-b026-ca105a6eca86-VaultNode/i-06f09e6fc27076955"
]
}
}
}
```

Am I missing something? How do I allow the HCP Vault user to assume my lambda role? Any help would be appreciated. Thanks!