AWS – CIDR Not Within VPC CIDR Ranges

First, I must mention: you made a very grave error by duplicating subnets in your VPCs. Even if there is next to zero chance you'll need to route traffic between them, the RFC1918 address space is well large enough for you to give each VPC a unique subnet. I consult with a number of companies on AWS topics, and I maintain a "master subnet list" spreadsheet to record subnets in use as I allocate VPCs for customers, to ensure I don't have overlapping subnets.

The obvious answer to your question is to re-number your overlapping VPCs. This is going to be painful, but it's the right answer to this problem, and will solve this issue for you once and for all.

If that's not an option, I can think of a couple of other options:

1. Utilize SQS for your logs - send logs from your app VPCs to an SQS queue, annotated with a source ID for each of your app VPCs, and then consume the logs out of SQS from your logging VPC. In addition to solving your stated problem, this puts a very highly-available buffer between your logs producers and log consumers. This protects you from losing logs if your infrastructure hiccups, or if you need to take it down for maintenance.
2. Expose your logging endpoint via a public IP (ELB, EIP, etc.), firewall it so that only your app servers' public IPs can hit it and have them send their logs this way. The traffic will remain on AWS's network, and as long as it's encrypted and authenticated, it's not much of a security issue. You'll pay more for bandwidth, though.

General Rule

NAT provides outgoing internet access, and does not allow incoming connections. Note that NAT gateway is a managed service, and is generally preferred over a NAT instance.

RDS does not need to initiate connections to the internet, so it does not need access to a NAT instance / gateway. RDS is fully managed by AWS, including software updates of database that runs RDS (MySQL, PostgreSQL, etc).

RDS Replication

With database replication internet ingress is required, but that isn't provided by NAT.

RDS Exposed to the Internet

You can expose an RDS instance to the internet (aka "make it publicly accessible"), but that's incoming rather than outgoing so NAT isn't required.