requestTemplates and responseTemplates not working in AWS API Gateway SAM template - aws-api-gateway

Question

I am having a problem getting the requestTemplates and responseTemplates I define to work.
The relevant piece of my (processed) template is here:
/cse:
options:
x-amazon-apigateway-integration:
type: mock
requestTemplates:
application/json: |
{
"statusCode" : 200
}
responses:
default:
statusCode: '200'
responseTemplates:
application/json: |
{}
responseParameters:
method.response.header.Access-Control-Allow-Origin: '''*'''
method.response.header.Access-Control-Allow-Methods: '''GET,OPTIONS'''
summary: CORS support
responses:
'200':
headers:
Access-Control-Allow-Origin:
schema:
type: string
Access-Control-Allow-Methods:
schema:
type: string
description: Default response for CORS method
get:
x-amazon-apigateway-integration:
httpMethod: POST
requestTemplates:
application/json: |
{
"body": $input.json('$'),
"headers": {
#foreach($header in $input.params().header.keySet())
"$header": "$util.escapeJavaScript($input.params().header.get($header))" #if($foreach.hasNext),#end
#end
},
"method": "$context.httpMethod",
"params": {
#foreach($param in $input.params().path.keySet())
"$param": "$util.escapeJavaScript($input.params().path.get($param))" #if($foreach.hasNext),#end
#end
},
"query": {
#foreach($queryParam in $input.params().querystring.keySet())
"$queryParam": "$util.escapeJavaScript($input.params().querystring.get($queryParam))" #if($foreach.hasNext),#end
#end
}
}
type: aws
responses:
default:
statusCode: 200
responseTemplates:
text/html: $input.path('$')
responseParameters:
method.response.header.Access-Control-Allow-Origin: '''*'''
method.response.header.Access-Control-Allow-Methods: '''GET,OPTIONS'''
uri: !Sub >-
arn:aws:apigateway:${AWS::Region}:lambda:path/2015-03-31/functions/${CSELambda.Arn}/invocations
responses:
'200':
headers:
Access-Control-Allow-Origin:
type: string
Access-Control-Allow-Methods:
type: string
description: OK
The options bit is auto-generated by AWS by adding a Cors entry in the template and works fine.
See how it defines a responseTemplates and requestTemplates and it works.
The get bit is the one I am trying to get to work. As far as I can tell, I am following the same format as the auto-generated options bit, but the API Gateway is not setting these 2 templates, and I am at a loss.
Been Googling, trying different things to no avail, so far. Any help appreciated.