Not the easiest way, but you can use TAPIR, or EPTE to convert binary to .TAP, and after load it into EP, like using tape.

Is there any easy way to run a binary rom the PC on the Enterprise? Via the serial port or something?

I've written about this, that SD card solution is great but not for everything, thus an Ethernet solution would be really superb. On the other hand, a simple serial connection is much faster solution to develop and you don't even need any additional code/program to do it, you can LOAD via the SERIAL: device as far as I know at least. The only problem is EP's "odd" serial interface; it is not really RS232 compatible at signal levels ... EP provides a +5V pin on the serial connector as "reference voltage" and it should be treated as the "offset ground" so this way you will have +/- voltages acceptable for standrad serial specification. However, treating +5V as ground can cause odd loops which can even cause short circuit etc. What I thought, that with some soldering an std TTL level connection can be done (so 0 and +5V relative to GND, maybe it's enough to "avoid" the op-amp part in input pins and the pull-up resistors to +12V on the output or so - but schematics of EP can be checked out more carefully to see this) and then it's easy to interface with eg a MAX232 IC which converts TTL levels to serial voltage specification and vice versa. The "offset ground" soltion feels a bit dangerous for me because of the reasons I've mentioned above in this post. But since maybe I am not the most experienced guy here on EP hardware, I can be wrong of course.

What is the problem with your interface?

I think use floppy disks are the most easy way.

When I plug it in, nothing seems to happen! I'm sure it used to work, but I think it was unrealiable. I'll have to take it apart and see what's happening...

Does it need an external power supply to the bus bridge? I don't remember having to use one.