I think that it might be worth investing a few dollars to try a usb y-cable such as
this one, connecting the two larger connectors to both of the BDP-1's rear ports. If the two rear ports are independently powered (you might want to ask James to confirm that), doing so would double the amount of power that can be provided to your hard drive. I'm still suspicious about the possibility that the drive may be marginally powered, even though it supposedly should work when connected to a rear port.
Alternatively, if you have one readily available you might try connecting the drive via a powered usb hub.
I notice that the descriptions of the Iomega SSD drive Cmo is using indicate that it uses significantly less power than comparable mechanical drives.
Regards,
-- Al