AWS Networking
AWS networking is not transitive. You generally can't hop between networks without making an explicit provision for it. You have to have a target in the VPC you're connected to, and from that target you can connect out to anything peered.
If your management VPC is peered with your VPC1 and VPC2 then you can't access VPC 1 / 2 directly from your laptop with Client VPN. To get access you'll need to put a bastion server or similar service (AWS Appstream, AWS workspace, etc) into the management VPC.
Suggestion - Multi-Account Architecture
This kind of setup is more commonly done using a multi-account architecture, which provides for additional isolation.
You could set up AWS Control Tower as a best practice environment to get you started, then set up AWS Transit Gateway in a dedicated networking account or the management account for communications between the accounts / VPCs. Client VPN can be set up in the same account as transit gateway and can provide access to all accounts / VPCs if configured correctly, but that's more of a security risk - a bastion helps with isolation.
You might get additional control by putting a NAT gateway in an account if you don't need public internet ingress, or by putting in squid proxies to limit what sites your servers can connect to - OS update sites for example.
This will probably cost more than your current setup. It's more of an enterprise strategy, where security and compliance is more important than total cost.