You should be able to dump the BIOS within DEBUG.COM (from DOS).
Ranges on a Rainbow-100 A are:
ROM (FA000-FBFFF) (E89) 8 K
ROM (FC000-FDFFF) (E90) 8 K
ROM (FE000-FFFFF) (E91) 8 K
The character ROM [E98] 2732 (4 K) can't be accessed this way.
Unverified:
C:\> DEBUG
-N DEC.BIN (resulting file will be named DEC.BIN)
-R BX (set BX=0000H/CX=6000H as count of bytes to write, 00006000H = 24K)
BX 0000
:0000
-R CX
CX 0000
:6000
-M FA00:0 6000 0100 (copy 24K from FA00:0 to offset 0100 in local segment)
-W 0100 (write from offset 0100 in local segment)
Writing 6000 bytes
-Q
Derived from instructions for IBM-PC computers -
http://www.mess.org/dumping/dump_bios_using_debug