My earlier Mac software is a little rusty since I haven't been using it much any more, but a couple of the later Mac OSes have a control panel that allows you to mount the Prodos volumes onto the desktop. I know Mac OS 7.6.1 has this control panel, but am unsure of OS 7.5.5. But this still requires having a disk image open to copy the files over.
The IIe card cannot make use of disk images such as .dsk, .po, .2mg, so to get the files out of these disk images, CiderPress is the best at that. CiderPress can also handle a .img disk image which can then be opened on the old Mac using DiskCopy.
So, for me anyways, the simplest solution at this point is to use CiderPress to create a 10 Mb disk image (or 32 Mb) that is DiskCopy compatible and copy some boot up files to it. The DiskCopy image has a suffix of .img
These are the files that you should have on it.
Prodos (v1.9)
Basic.system (v1.4.1)
DiskMaker8 - to create disk images
Dsk2File58 - open .dsk, .do and .po disk images and copy files to a volume
This setup will work on an unenhanced IIe. Copy this disk image to a cd and transfer to your old Mac. Use Disk Copy to open the disk image which will mount on the desktop. Set up the IIe card to boot from this mounted disk image.
Now, with the use of the prodos control panel you can copy all the disk images on the other CD to one of the Prodos volumes, you can use Dsk2File58 to open those disk images and copy the files to your Prodos volumes.
Hope these instructions are clearer than mud. If don't have a PC to use CiderPress, email me and I can email you a Disk Copy compatible disk image with the required files.