AWS Serverless MySQL/Postgres offerings are straight trash. I used v1 to build a new app but had nothing but problems. Extremely slow starts (from zero), horrible scaling (it would always get stuck), (relatively) huge bills for the smallest capacity, limitations all over the place. After the first year on that I looked into v2 but my costs would have doubled and I didn't believe their promises of faster scaling. I moved to PlanetScale [0] and was very happy ($30/mo covered prod and dev/qa vs well over that for v1 even with having scale to zero on the AWS dev/qa instances). Also you can quickly be forced into paying for RDS Proxy if you are using lambdas/similar which is not cheap (for me). PS doesn't scale to 0 but at the time $30/mo was a decent savings over AWS Aurora Serverless.
This year I started to run into some issue with PS mainly around their plans changing (went from pay for reads/writes/storage to pay for compute/storage). Yes, yes, I know they still offer the $30/mo plan but it's billed as "Read/write-based billing for lower-traffic applications" and they dropped all mentions of auto-scaling. That coupled with them sleeping your non-prod DB branches (no auto-wakeup, you had to use the API or console) even after saying that was a feature of the original $30 plan rubbed me the wrong way. Eventually the costs (for what I was getting) were way too out of whack. My app is single-tenant (love it or hate it, it's what it is) so for each customer I was paying $30/mo even though this is event-based software (like in-person, physical events that happen once a year) so for most the year the DB sat there and did nothing.
Given all that I looked into Neon [1] (which I had heard of here on HN, but PS support suggested them, kudos to them for recommending a competitor, I always liked their support/staff) and while going from MySQL to Postgres wasn't painless it was way easier than I had anticipated. It was one of the few times Prisma "just worked", I don't think I'd use it again though, that DB engine is so heavy especially in a lambda. I just switched over fully last week to Neon and things seem to have gone smoothly. I can now run multiple databases on the same shared compute and it scales to 0. In fact it's scale up time is absurdly fast, the DB will "wake up" on it's own when you connect to it and unlike AWS Aurora Serverless v1 it comes up in seconds instead of 30-60+ so you don't even have to account for it. With AWS I had to have something poll the backend waiting to see if the DB was awake yet, to fire off my requests, if it was asleep. With Neon I don't even consider it, the first requests just take an extra second or two if that.
I don't have any ill will towards PlanetScale and I quite enjoyed their product for almost the whole time I used it. Also their support is very responsive and I loved the branching/merging features (I'll miss those but zero-downtime migrations aren't required for my use-case, just nice to have). In fact if I had written my app to be multi-tenant then I'd probably still be on them since I could just scale up to one of their higher plans. It does seem like Neon is significantly (for me/my workload) cheaper for more compute, I had queries taking _forever_ on PS that come back in a second or less on Neon all while paying less.
All that said, I _highly_ recommend checking out Neon if you need "serverless" hosting for Postgres that scales to 0.
[0] https://planetscale.com/
[1] https://neon.tech/