VLAN popping, pushing and/or swapping operation

Topology:

Office HQ <--------> (ge-1/1/4) core1 (ge-1/1/0) <-------> (ge-1/1/0) core2 (ge-1/1/3) <---------> customer router

Office HQ        - VLAN 600

core1.ge-1/1/4 - Outer VLAN 1, Inner VLAN 600

core1.ge-1/1/0 - VLAN 336

core2.ge-1/1/0 - VLAN 336

core2.ge-1/1/3 - Outer VLAN 336, Inner VLAN 600

customer router - VLAN 600

core2.ge-1/1/3 and core2.ge-1/1/0 are bridged together.

core1.ge-1/1/4 and core1.ge-1/1/0 are bridge together.

core1.ge-1/1/0 learns the customer routers MAC address.

How can Office HQ communicate with customer router over VLAN 600? Does there need to be some swap operation on core1.ge-1/1/0 to change VLAN 1 to 336 and the other way around?

Configurations:

core1.ge-1/1/4:

unit 5103 {
    description "SPH management QinQ";
    encapsulation vlan-bridge;
    vlan-tags outer 0x8100.1 inner 0x8100.600;
}

core1.ge-1/1/0

unit 100 {
    encapsulation vlan-bridge;
    vlan-id 336;
}

bridge-domain on core1:

domain-type bridge;
    interface ge-1/1/0.100;
    interface ge-1/1/4.5104;

core2.ge-1/1/0

unit 100 {
    encapsulation vlan-bridge;
    vlan-id 336;
}

core2.ge-1/1/3

unit 336 {
    description "SPH management VLAN 600";
    encapsulation vlan-bridge;
    vlan-tags outer 0x8100.336 inner 0x8100.600;
} 

 bridge-domain on core2:

domain-type bridge;
    interface ge-1/1/3.336;
    interface ge-1/1/0.100;