Getting your live stack to call your Dev machine seems a sure fire way to slip up and accidentally send customers a test email. Or accidentally order $100,000,000 widgets.
In fact, I would say it sounds great, but in fact it's incredibly risky and monumentally stupid.
Just create a staging env, if you don't have it already, and point to staging
When you write "your live stack", my brain freezes. Even a simple "webapp" project in my team had 4 common stacks (prod, staging, dev, git-master which had every push to master getting deployed) plus individual stacks, one per developer, with the ability to create extra stacks for specific feature development.
Only prod had real data. Obviously (to me) you wouldn't shorcircuit lambdas in the prod env. Not even staging. Not even dev. Not even git-master. Short-circuit development environments where you explore. Very low risks there. We need to learn it's ok to fuck up, as long as we fuck up in a safe environment, nowhere near the nuclear button ;)
I did wonder with regards to the IP address. If you had a VPN set up for access to a VPC and ran the lambda in that VPC then that would be another option I think. You would just need to configure a static IP for your dev machine within the private network.