You may consider using NAT instance of EC2. A micro instance which can also serve as your bastion host.
Or if you are a true extreme penny pincher -- have your lamba function invoke aws api to set up a NAT Gateway and update the subnet route, then execute your business function and then clean up the NAT.
Huh? I can’t make out if this is sarcasm... you suggesting opening the firewall from the inside for each request, finishing the request and then closing the firewall? For starters, what would happen if request 1 closed the firewall while request 2 was still working?
Or if you are a true extreme penny pincher -- have your lamba function invoke aws api to set up a NAT Gateway and update the subnet route, then execute your business function and then clean up the NAT.