It would have been easier if you have just provided the data as binary files.
Anyway, it appears that your CMOS RAM has a size of 128 bytes, not 256. Bytes 128 to 255 appear to be a mirror image of bytes 0 to 127.
The original IBM PC/AT had 64 bytes of CMOS RAM. Later chipsets (eg Chips & Tech 286) had "extended" CMOS RAM which occupied 128 bytes.
Those bytes which you have identified as differing between successive reads are in fact related to the time (byte 0 = seconds, 2 = minutes) and are therefore variable. They are not included in the checksum calculation.
In fact here is the address map:
http://stanislavs.org/helppc/cmos_ram.htmlBytes 0x2E and 0x2F constitute a 2-byte checksum which is calculated over addresses 0x10 to 0x2D (not 0x20 as indicated at the previous URL).
For an AMI 386 BIOS, bytes 0x3E and 0x3F store a 2-byte extended CMOS checksum which is computed over addresses 0x34 to 0x3D and 0x40 to 0x7F. Your BIOS appears to be different in this respect.
Byte 0x37 stores a password encryption "key". Bytes 0x38 to 0x3D store an encrypted password. Once again, your BIOS appears to be different in this respect.
What you need to do is to make just one minor change to your BIOS configuration such that this change is recorded in your CMOS RAM. Then compare the CMOS RAM dumps before and after the change. Depending on the type of change, you may find that it is written to flash memory rather than to CMOS RAM.
You say that you have "found the relevant bit", so which bit is it?
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