With a bit of assistance, DSolve can obtain the desired solution. ode has a first integral, which is obtained by

Integrate[Subtract @@ ode, x, GeneratedParameters -> C] == 0
(* -E^y[x] + C[1] + Derivative[1][y][x] == 0 *)

Applying the initial conditions then yields

Solve[% /. x -> 3 /. Rule @@@ ic, C[1]] // Flatten
(* {C[1] -> 0} *)

DSolve now obtains the desired solution

DSolve[{%% /. %, ic}, y[x], x] // Flatten
(* {y[x] -> -Log[4 - x]} *)

With a bit of assistance, DSolve can obtain the desired solution. ode has a first integral, which is obtained by

Integrate[Subtract @@ ode, x, GeneratedParameters -> C] == 0
(* -E^y[x] + C[1] + Derivative[1][y][x] == 0 *)

Applying the initial conditions then yields

Solve[% /. x -> 3 /. Rule @@@ ic, C[1]] // Flatten
(* {C[1] -> 0} *)

DSolve now obtains the desired solution

DSolve[{%% /. %, ic}, y[x], x] // Flatten
(* {y[x] -> -Log[4 - x]} *)

Addendum

To answer why DSolve fails in the question, consider

s = DSolveValue[{ode, First@ic}, y[x], x, , Assumptions -> C[1] ∈ Reals]
(* -Log[-(1/C[1]) + (E^(3 C[1] - x C[1]) (1 + C[1]))/C[1]] *)

Applying the second boundary condition then yields

Solve[1 == D[s, x] /. x -> 3, C[1]] // Flatten
{C[1] -> 0}

Of course, inserting this value of C[1] into s fails.

Power::infy: Infinite expression 1/0 encountered.

On the other hand, as suggested by Mariusz Iwaniuk in a comment,

Limit[s, %]
(* -Log[4 - x] *)

does work. Apparently, DSolve does not try this.