My attempt is modeled on the this tutorial.
I am able to ping from the namespace to the network if the physical interface is not assigned to the bridge.
# Create namespace
ip netns add namespace1
# Create veth pair.
ip link add veth1 type veth peer name br-veth1
# Associate the non `br-` side with the namespace.
ip link set veth1 netns namespace1
# Give namespace-side veth ip addresses.
ip netns exec namespace1 ip addr add 192.168.1.11/24 dev veth1
# Create a bridge device naming it `br1` and set it up.
ip link add name br1 type bridge
# Turn up the bridge.
ip link set br1 up
# Set the bridge veth from the default namespace up.
ip link set br-veth1 up
# Set the veth from the namespace up too.
ip netns exec namespace1 ip link set veth1 up
# Add the br-veth1 interface to the bridge by setting the bridge device as their master.
ip link set br-veth1 master br1
# Add the physical interface to the bridge
ip link set enp3s0 master br1
# Set the address of the `br1` interface (bridge device) to 192.168.1.10/24 and also set the broadcast address to 192.168.1.255 (the `+` symbol sets the host bits to 255).
ip addr add 192.168.1.10/24 brd + dev br1
# add the default gateway in all the network namespace.
ip netns exec namespace1 ip route add default via 192.168.1.10
# Set us up to have responses from the network.
# -t specifies the table to which the commands should be directed to. By default, it's `filter`.
# -A specifies that we're appending a rule to the chain that we tell the name after it.
# -s specifies a source address (with a mask in this case).
# -j specifies the target to jump to (what action to take).
iptables -t nat -A POSTROUTING -s 192.168.1.0/24 -j MASQUERADE
sysctl -w net.ipv4.ip_forward=1