Could this "symmetric RSA" scheme provide key compromise resistant communications?
Yes, this seems to make sense and it is a plausible solution. The KEM approach does not work, unless you use some tricks, like including a hash of the message in the KEM. (That could work, of course.)

Security goal

The type of scheme we are looking at consists of three algorithms $(K,E,D)$. The key generation algorithm $K$ outputs two keys, say $k_0$ and $k_1$. The encryption algorithm $E$ takes as input a key and a message and outputs a ciphertext. The decryption algorithm $D$ takes as input a key and a ciphertext and outputs a message or $\bot$.

We require that for any $(k_0, k_1)$ output by $K$ and any message $m$, $$D(k_{1-\beta}, E(k_\beta, m)) = m \text.$$

There are two goals here, confidentiality and integrity. You want messages encrypted with $k_\beta$ to be confidential to everyone who does not have $k_{1-\beta}$. And someone who does not have $k_\beta$ (but may have $k_{1-\beta})$ should not be able to create ciphertexts that look as if they were created with $k_{\beta}$.

Confidentiality

In the confidentiality game, the simulator generates a key pair $(k_0, k_1)$ and gives $k_1$ to the adversary. The adversary chooses two messages $m_0$ and $m_1$. The simulator flips a coin $b$ and creates a ciphertext $c^*$ as $E(k_1, m_b)$. The simulator sends $c^*$ to the adversary, and the adversary eventually sends $b'$ to the simulator. During the entire process, the adversary is allowed to send messages or ciphertexts to the simulator, and the simulator responds with the result of $E(k_0, m)$ or $D(k_0, c)$, respectively. 

The adversary wins if $b=b'$. He should not win significantly more (or less) often than half the time.

Integrity

In the integrity game, the simulator generates a key pair $(k_0, k_1)$ and gives $k_1$ to the adversary. The adversary is allowed to send ciphertexts and messages to the simulator, and the simulator responds with the result of $E(k_0, m)$ or $D(k_0, c)$, respectively. Finally, the adversary sends a ciphertext $c^*$ to the simulator and the simulator decrypts it using $k_1$.

The adversary wins if the $c^*$ decrypts correctly and it was not created by the simulator. The adversary's success rate should be negligible.

Analysis Proposal

I would try something like OAEP+, but for simplicity, I would "personalize" the hash functions so that we can use different random oracles for the two senders. The confidentiality game looks a lot like the usual RSA-OAEP+ game, but you need to create ciphertexts for the recipient. This becomes much easier when you use separate random oracles.

I suspect it will work out. It may even be possible to "reduce" the security to that of ordinary RSA-OAEP+.