Determining if a TM decidable or not, with limited information
A TM cannot be "decidable". What you are asking is whether the language that you describe is decidable.  

Hint: the language $A_{TM}=\{&lt;M,w&gt;: M \text{ accepts }w\}$ is undecidable even if we restrict to TMs with a binary work alphabet. Try to reduce from this problem, by utilizing only the letters $\{a,B\}$ from $\Gamma$. Then, you can use $b$ only when you need it.

Please comment if you need further hints.