Here is your problem. You are trying to build a huge application using inadequate technical building blocks.
Lambdas are super inefficient in many different ways. It is a good tool but as with every tool you do need to know how to use it. If you try to build heavy compute app in Python and then complain at your electricity bill -- that really is on you.
If your database is overloaded with hundreds of thousands of connections from your lambdas, it means it is end of the road for your lambdas. Do not put effort into scaling your database up, put effort into reducing the number of your connections and efficiency of your application.