I think a mixture of persistent proxies could work well with lambda functions for this use case. You can run lambdas in your VPCs with assigned security groups

I think a mixture of persistent proxies could work well with lambda functions for this use case. You can run lambdas in your VPCs with assigned security groups and IAM roles just like an EC2 instance and therefore lambda functions should be able to proxy through an intermediate server that performs the connection pooling on behalf of calling functions. Not sure what the standard solution for something like this would be though because every time I've seen connection pooling it's for use locally. Perhaps some nginx or haproxy hacking could make it work.