You can use Cognito with a "public" pool id, then attach role to the Cognito pool id, the role being accessing your Lambda, I think it is called InvokeLambdaRole or something
AWS.config.credentials = new AWS.CognitoIdentityCredentials({
IdentityPoolId: 'REGION:YOUR_POOL_ID',
});
Use AWS STS to get temporary credentials with limited privileges. After that you can use API Gateway with AWS_IAM authentication, then end point will invoke the Lambda methods for you. Or you can invoke lambda directly with the credentials you got, but then again you have to attache the right roles for the identity pool you created.
NB: Put strictly minimum roles on your pole, that is a publicly available id, every body can use it to get a temporary or a fixed (to track users accross devices) user_/app_ id.