The Card featured 256 bytes of ROM.
Is there evidence documenting how the cassette program was stored?
The evidence is right there in the PCB photo you added. The two MMI 6301 chips, labled APPLE A3 and APPLE A4, at position 3 and 4, are 256 by 4 PROMs. The same type, produced by Monolithic Memories, as used on the Apple 1 motherboard for the monitor ...
Both the DFS and the ADFS did indeed reserve memory for themselves at startup, increasing PAGE; this memory was used to hold the current catalogue and other bookkeeping.
For adapting existing tape software two observations are helpful:
almost nothing on the BBC was a multiload; and
you can turf the [A]DFS out of memory if you don't intend to use it again.