The nice examples given by Pace show that the answer is negative in general. There is also a natural condition on $R$ which gives a positive solution. It involves the Jacobson radical of a non-unital ring. (I'm following the first few exercises in section 4 of Lam's A First Course in Noncommutative Rings.)
The binary operation $a \circ b = a + b -ab$ on $R$ is associative and has unit element $0$. An element of $R$ is called left (right) quasi-regular if it has a left (right) inverse in the monoid $(R, \circ, 0)$ and is called quasi-regular if it has both a left and a right inverse, which are necessarily equal.
The Jacobson radical $J(R)$ is the sum of all left ideals of $R$ consisting of left quasi-regular elements. One can show that this is an ideal of $R$, and all elements in it are quasi-regular. So $J(R)$ is the largest left (right) ideal consisting of left (right) quasi-regular elements. One can show that this coincides with the usual Jacobson radical in case $R$ has an identity.
Claim: For a (not necessarily unital) ring $R$, the following are equivalent:
- $R$ is the unique maximal right (or left) ideal of its standard unitization $R^\sharp$;
- $R = J(R)$.
Proof: Assume 1 holds, so that $R^\sharp$ is local and $R = J(R^\sharp)$. Then for any $x \in R$, the element $1 - x = 1 \oplus (-x) \in K \oplus R = R^\sharp$ lies in $R^\sharp \setminus R$ and therefore is invertible. Its inverse is of the form $\alpha \oplus y \in R^\sharp$. Writing out the equation for $(1-x)(\alpha \oplus y) = 1 \oplus 0$ yields that $\alpha = 1$ and $x + y - xy = 0$, and similarly we obtain $y + x -yx = 0$. So every element of $R$ is quasi-regular and 2 holds.
Conversely, assume 2 holds. We know that $R$ is a maximal right (and maximal left) ideal of $R^\sharp$ simply because of the ring isomorphism $R/R^\sharp \cong K$; thus $J(R^\sharp) \subseteq R$. If we can show that $R$ is contained in $J(R^\sharp)$ then we will be done, for then $R \subseteq J(R^\sharp) \subseteq R$ yields that $R^\sharp/J(R^\sharp) = R^\sharp/R \cong K$ making $R$ local with unique maximal right ideal $R$.
So let $x \in R$. To show $0 \oplus x \in J(R^\sharp)$, let $\alpha \oplus r \in R$; we need to find a left inverse to $$1 \oplus 0 - (\alpha \oplus r)(0 \oplus x) = 1 \oplus-(\alpha x + rx) = 1 \oplus (-x')$$ where we set $x' = \alpha x + rx \in R$. To this end, fix a left quasi-inverse $y$ for $x$ in $R$. Then from $y+x'-yx' = 0$ we obtain $$(1 \oplus -y)(1 \oplus -x') = 1 \oplus 0$$ as desired. QED