+1 to ProxySQL, especially in RDS environments with huge monoliths attached that open a shitload of threads. RDS has fixed max_connections depending on the instance size so if you don't want to pay $$$$ for bigger but underused instances - and while you are trying to get the devs update all the hundreds old dependencies in the monolith to improve it, ProxySQL - can save your day. It did, for me. And yes, it's a self-managed system but it's pretty easy to operate and very stable.

You can change the max_connections in RDS though. The default is insanely high and I have no idea what it is that way. 4vCPU instances running with 5k max connections iirc, I have never seen an instance this size handle more than 100-200 concurrent connections on a CPU bound workload.