AWS EC2 lost connection to SSM after input tailscale set --exit-node=x.x.x.x

Hello,

I am having an issue when I setup the tailscale in my AWS env.

I have two ec2 in two different VPC which only have private IPs (A and B). I setup A as client and B as the exit node. Follow the instruction to install tailscale and configure them separately.

Everything looks good from the web portal of tailscale. Both two nodes are connected and get IPs (100.X.X.X).

When I type `tailscale set --exit-node=100.x.x.x `on the client A, then I lost the connection to this ec2. From AWS, it shows the warning below:
Instance is not connected to Session Manager
To connect to an instance using Session Manager, the instance must be able to connect to the Systems Manager endpoints.

I think this is because the routes got changed all of the traffic is forwarded to the exit node. So, the ec2 lost the connection to the Systems Manager endpoints.

Can anyone help to fix this issue? I still want to route the traffic going to the Internet through the exit node, but at least the traffic to the Systems Manager endpoints can go through the original gateway.

Thanks,