Note that we have $$a + b = (a \oplus b) + ((a \land b) \ll 1). \tag{1}$$ So asking whether $a + b = a +_K b$ is asking when $x \oplus y = x + y$ where $x = a \oplus b$ and $y = ((a\land b) \ll 1)$. On the other hand by the same expansion as (1), we have $x \oplus y = x + y$ if and only if $x \land y = 0$.
Let us draw $a, b$ uniformly at random from the set of $n$ bit non-negative integers. This is the same as picking each bit of $a$ and $b$ independently at random (with uniform probability over $1$ and $0$).
We have $\Pr(x\land y \not=0) = \Pr(\exists i x_i \land y_i = 1)$, and expanding $x_i, y_i$ this is equal to $\Pr(\exists i (a_i \oplus b_i) \land a_{i-1} \land b_{i-1})$. We can now lower bound this probability, by $\Pr(\exists i \textrm{ even}, (a_i \oplus b_i) \land a_{i-1} \land b_{i-1} = 1) = 1 - (7/8)^{n/2}$ since those events are independent for different even $i$.
To provide a more precise estimate, let us consider a three-state Markov which is processing bits $a_{i}, b_{i}$ in the order of decreasing $i$, and keeps as its state $a_{i} \oplus b_{i}$.
I.e. we have three states $0, 1, F$, and the transition matrix is given by $$A = \left(\begin{matrix} 1/2 & 1/2 & 0 \\ 1/4 & 1/2 & 1/4 \\ 0 & 0 & 1 \end{matrix}\right)$$ That is, if we are in the $F$ state, we stay in the $F$ state, if we are in the $0$ state we move to state $0$ or $1$ with probability $1/2$, if we are in the $1$ state we have probability $1/4$ of moving to the $1$ state and $1/4$ of moving to the $F$ state.
We are starting in the state $0$, and are interested in what is the probability of being in the state $F$ after $n$ steps. That quantity is given by $e_1^T A^n e_H$.
Using some standard tool (I used numpy and python), we can diagonalize $A = U \Sigma U^{-1}$, where $$U \approx \left(\begin{matrix} 0.81649658 & -0.81649658 & 0.57735027 \\ 0.57735027 & 0.57735027 & 0.57735027 \\ 0. & 0. & 0.57735027\end{matrix}\right)$$ and $$\Sigma \approx \left(\begin{matrix} 0.85355339 & 0 & 0 \\ 0 & 0.14644661 & 0 \\ 0 & 0 & 1 \end{matrix}\right)$$
Hence $e_1^T A^n e_3 = (e_1^T U) \Sigma^n (U^{-1} e_3) = \alpha_1 \sigma_1^n + \alpha_2 \sigma_2^n + \alpha_3 \sigma_3^n$ where $\alpha \approx (-1.20710678, 0.20710678, 1.)$ and $\sigma \approx (0.85355339, 0.14644661, 1.)$.
I.e. the failure probability looks a bit like $1 - 1.2 \cdot 0.8^n + 0.2 \cdot 0.15^n$.