Normally a bridge is expected to be fully transparent at L2 (data link layer), so it will carry Ethernet frames unaltered – whenever your router sends a frame, the ISP will see your router's WAN interface MAC address. (Compare: Ethernet switches are bridges.)
So when the DSL modem is in bridge mode, it will not interact with the ISP at Ethernet level on its own; it will especially not interact at IP level and will not try to obtain a WAN address; and so its MAC address won't be seen by the DHCP server.
This isn't necessarily true for all WAN technologies (some of which are unable to preserve the MAC address and might resort to pseudo-bridging aka 'arpnat' mode; e.g. this is done when a Wi-Fi client tries to act as a bridge.) However, as far as I know, it is always true for ADSL IPoE/PPPoE and especially VDSL, both of which carry standard Ethernet-shaped data.