Jelly, 4 bytes

HL\^

A monadic Link that accepts an integer and yields a list of one integer. ...Or a full program that accepts an integer and prints the result.

The code's bytes only change by one bit each time, as required:

Try it online!

How?

The shortest code that starts with H (halve) and ends with ^ (XOR the input) for which the code in between has no adverse effect while fulfilling the source-code restriction of only changing by at most one bit from byte to byte.

HL\^ - Link: integer, N
H    - halve N                    -> N/2
  \  - cumulative reduce with:
 L   -   length                   -> [N/2]
   ^ - (floor to integer) XOR (N) -> [floor(N/2) XOR N]

Jelly, 4 bytes

HL\^

A monadic Link that accepts an integer and yields a list of one integer. ...Or a full program that accepts an integer and prints the result.

The code's bytes only change by one bit each time, as required:

Try it online!

How?

The shortest code that starts with H (halve) and ends with ^ (XOR the input) for which the code in between has no adverse effect while fulfilling the source-code restriction of only changing by at most one bit from byte to byte.

HL\^ - Link: integer, N
H    - halve N                    -> N/2
  \  - cumulative reduce with:
 L   -   length                   -> [N/2]
   ^ - (floor to integer) XOR (N) -> [floor(N/2) XOR N]

Note

:2 (integer divide by two) seems like a reasonable start as the bytes already only differ by one bit, but the next available bytes (Ø"036:rḶ) don't really help us advance.