"Invalid API identifier" when referencing AWS::Serverless::Api resource ID

I have a CloudFormation template managing a series of nested templates. One of these templates is a SAM application, which creates a AWS::Serverless::Api resource and outputs this resource reference. I am trying to add another route to this API in a separate nested template (or the parent template, but I get the same error here). My setup looks like this (simplified):

# parent template
AWSTemplateFormatVersion: 2010-09-09

Resources:
  ContactFormApi:
    Type: AWS::CloudFormation::Stack
    Properties:
      TemplateURL: ./node_modules/@dawaltconley/contact-api/dist/build/template.yaml
  MailingList:
    Type: AWS::CloudFormation::Stack
    DependsOn:
      - ContactFormApi
    Properties:
      TemplateURL: ./mailing-list.template.yaml
      Parameters:
        ApiGatewayId: !GetAtt ContactFormApi.Outputs.ApiId

# ContactFormApi template
AWSTemplateFormatVersion: '2010-09-09'
Transform: AWS::Serverless-2016-10-31

Resources:
  ContactApi:
    Type: AWS::Serverless::Api
    Properties:
      StageName: Prod

Outputs:
  ApiId:
    Value: !Ref ContactApi

# MailingList template
AWSTemplateFormatVersion: 2010-09-09

Parameters:
  ApiGatewayId:
    Type: String

Resources:
  ApiIntegration:
    Type: AWS::ApiGatewayV2::Integration
    Properties:
      ApiId: !Ref ApiGatewayId
      IntegrationMethod: GET
      IntegrationType: AWS_PROXY
      IntegrationUri: !GetAtt LambdaFunction.Arn

  ApiRoute:
    Type: AWS::ApiGatewayV2::Route
    DependsOn:
      - LambdaFunction
      - ApiIntegration
    Properties:
      ApiId: !Ref ApiGatewayId
      RouteKey: GET /subscribe
      Target: !Sub 'integrations/${ApiIntegration}'

  LambdaFunction:
    Type: AWS::Lambda::Function
    Properties:
      Code: directory
      Handler: index.lambdaHandler
      Runtime: nodejs18.x
      Timeout: 10
      Architectures:
        - x86_64
      Role: !GetAtt LambdaRole.Arn

  LambdaRole:
    Type: AWS::IAM::Role
    Properties:
      AssumeRolePolicyDocument:
        Version: '2012-10-17'
        Statement:
          - Effect: Allow
            Principal:
              Service: lambda.amazonaws.com
            Action: sts:AssumeRole

I am using aws cloudformation package and aws cloudformation deploy to deploy this template. When I do, it fails on the creation of the ApiIntegration resource, and gives the following error (I replaced my account number with 123456):

Invalid API identifier specified 123456:knyawz3heb (Service: AmazonApiGatewayV2; Status Code: 404; Error Code: NotFoundException; Request ID: cb5323c0-4cef-447b-9cbd-b86195321bca; Proxy: null)

I have confirmed that knyawz3heb is indeed the ID of the API Gateway resource I want to reference, that this is the value of the ApiGatewayId parameter that the AWS console shows for the failed MailingList template, I am pretty sure that this resource exists at the time that the ApiIntegration resource is being created.

I am wondering if there's anything I'm doing wrong here that I've totally missed? Is there a correct way to reference the API resource ID created by SAM? Is it possible that the Serverless transform is turning resource_id into account_number:resource_id? I tried turning it back with !Select [1, !Split [':', !Ref ApiGatewayId]], but this failed (no index 1).