I realise this is a couple of years old now, but for some reason appeared in my feed. There's a few issues with the article and some of the comments that I can clear up.

Most of it is covered in an article I wrote here on using API Gateway with an ALB to workaround the ugly and changing DNS issue. No it can't be solved just using CNAMES and Route53.

https://aws.plainenglish.io/custom-dns-for-private-api-gateway-5940cb4889a8

Also wanted to point out that 1 million free invocations is for the account, not per Lambda - don't want anyone getting caught out with that.

Other issues - you could always point 3 different methods on API Gateway to the same Lambda function if they can all be handled there; cold starts are just as much an issue behind an ALB as API Gateway; and cold starts would be less of an issue if you used Node or Python rather than Java/Sprinig boot.

Hope it all worked out for the team in the end!