Lambda unable to delete object from s3 in another account

I have a lambda function in a VPC in account A. The lambda is trying to delete an s3 object in account B. I created a VPC gateway endpoint in the VPC of account A.

I created a role in account A and added the policy below to it:

{
    "Action": "s3:DeleteObject",
    "Effect": "Allow",
    "Resource": "arn:aws:s3:::bucket-name-a/*"
},

I also added the policy below to the vpc endpoint:

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Effect": "Allow",
            "Principal": {
                "AWS": "arn:aws:iam::1111111:role/lambdaRoleA"
            },
            "Action": "s3:DeleteObject",
            "Resource": "arn:aws:s3:::bucket-name-a/*"
        }
    ]
}

I added the following policy to the s3 bucket:

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Sid": "AllowLambdaToDeleteS3Object",
            "Effect": "Allow",
            "Principal": {
                "AWS": "arn:aws:iam::1111111:role/lambdaRoleA"
            },
            "Action": "s3:DeleteObject",
            "Resource": "arn:aws:s3:::bucket-name/*"
        }
    ]
}

The lambda completes without an error, but when I check the bucket the object is still there.

However, when I change the policy in the vpc endpoint to Full access, the s3 object gets deleted.

I triple checked the spelling on everything, but can't seem to figure it out.

It appears that for gateway endpoints, the Principal element must be set to *. To specify the principal, you need to use the aws:PrincipalArn key like this:

"Condition": {
    "StringEquals": {
        "aws:PrincipalArn": "arn:aws:iam::1111111:role/lambdaRoleA"
    }
}

Just needed to add this inside of of the Statement at the same level as Resource.

Found the answer in the AWS documentation: https://docs.aws.amazon.com/vpc/latest/privatelink/vpc-endpoints-access.html

Tested it by added a different role initially - lambdaRoleB and the s3 object was not deleted. Later changed it to lambdaRoleA and the object was successfully deleted.