"By the way" in formal writing
No, I wouldn't recommend doing so, as the phrase is informal. There are many formal alternatives you could use to make you sound more professional, some of those are:
it is worth noting
it might also be noted/observed
The phrase by the way is not especially informal, and you may freely use it in formal situations. However, if you wish to use a variant which is more formal, then you could use a substitute such as:
- Speaking of which,
- This brings to mind
- Apropos
Or one of the phrases mentioned in Rimmer's answer.