Does every r.e. set containing the set of total recursive functions contain all partial recursive functions?
Take $A = \{i\ |\ \phi_i(0) \downarrow \}$. It is obviously r.e., and includes $\mathrm{Tot}$.

Yet, it does not contain any index of, say, the always-undefined function $u$. Nor indices for any recursive partial function which is undefined at $0$.

The Kleene set also works: $K = \{i\ |\ \phi_i(i) \downarrow \} \supset \mathrm{Tot}$, yet it does not contain any index for $u$. (And it is r.e.)