Should there be a mac address associated with this interface somewhere? Also should arp=proxy-arp be set on the bridges and the admin-mac address at both ends?
No and no. The bridges the BCP tunnels become dynamic ports of already have some MAC addresses; whether these are assigned automatically or using admin-mac has no impact on the operation of the BCP. Of course the bridges geting interconnected must not have identical MAC addresses, but this requirement is nothing BCP-specific.
arp=proxy-arp makes sense when you connect individual Windows machines as L3 (non-BCP) L2TP clients, as Windows automatically add a route via the tunnel to the whole A (/8), B (/12), or C (/24) class subnet their individually assigned address fits into. So placing the pool for these clients into the LAN subnet allows you to avoid additional configuration on Windows side, but you must make the local devices in the LAN send the packets for the Windows clients to the router, which is achieved by means of proxy-arp that makes the router respond with its own MAC address to ARP requests regarding the IP addresses of currently active L2TP clients.
The reason why I was asking whether you have assigned IP addresses to the tunnel endpoints was to find out whether that may be the reason why the ping reports duplicates. As said it was just a wild guess.
I can only imagine multi-path propagation between the two devices to cause such an effect, but if it was due to a plain L2 loop, it should either cause much more harm than just duplicating ICMP echo requests or responses if no loop prevention mechanism is in use, or it should only last until the loop prevention mechanism cuts the loop, so not longer than a few seconds. And it makes no sense that this would be different for the EoIP case and for the BCP case if the two are used, one at a time, to interconnect the same bridges.
The only mechanism known to me that intentionally sends the same packet through two distinct paths is bonding in broadcast mode, but you haven't mentioned anything like that. But you haven't mentioned a hundred other things about your configuration either, because they are presumably unrelated to the issue.
In fact, the problem with these mysteries is that they are always caused by something you assume to be unrelated. That's why it only helps to post the complete configuration export, not just those few bits you assume to be relevant.
If it was bonding in broadcast mode, it is still strange that it would only affect BCP and not EoIP.
Another strange thing is that the duplication seems to only happen in one direction, otherwise you should get 4 response packets to a single echo request packet (unless the ICMP stack remembers the serial number of the last request and doesn't respond duplicates - I'd have to investigate a bit to be sure but it seems to me it would be against the purpose of ICMP echo).
At this point I would use sniffing at both devices, matching only on the IP address of the remote ping participant and icmp, to see what exactly happens there - from where does the ping request come in, which interfaces it passes through, and at which device it actually takes two outgoing interfaces.