Putting directional mics close together only works well when they are angled apart enough from each other, which how they are usually setup. But if they aren't angled apart very much, they need to be placed further apart. And if both directional mics are pointing directly ahead, parallel to each other, they'll need the same spacing that omnis would.
We haven't heard back from obsidian, but I'll jump to what my suggestions will most likely initially be:
On the microphone configuration end of things, if the mics cannot be mounted any further apart, I'd suggest angling the mics further apart from each other if using directional mics like cardioids, or putting a baffle of some sort between them if they are omnidirectionals. A baffle can be anything that blocks sound from reaching the other side- his head, his body, a piece of cardboard, a Jecklin-disk a chair back, or whatever.
fwiw, I do not consider using a baffle as having the mics closely spaced, or even spaced at a distance. To me, that is more like running ambient and SBD, as they are physically separated.
Well you should. Consider this- what a baffle does is make an omnidirectional mic directional at high frequencies. The size of the baffle and the distance of the microphones from the baffle, affect the degree of directionality, the shape of that directionality, and the frequency at which that directionality becomes effective. As far as high frequencies are concerned, the mics are physically separated, but at low frequencies they are not physically separated. The low frequencies don't "see" the baffle at all, they go right around it as if it were invisible or not there at all.
Spacing between microphones matters when using a baffle just as it does any other time. In fact, in some ways getting the spacing right matters even more. Moke makes some classical recordings using a tiny square baffle with his miniature DPAs mounted almost right up against either side, very little spacing between the microphones and between the microphones and baffle. It's very small and very compact. And that might something obsidian can try. At the opposite extreme, Ray Kimbers gigantic ISOphone system (I think that's what it's called) uses a giant baffle that needs a crane to lift it, and uses a mic spacing of something like 8-10' or more. Over the course of his career, Jorg Jecklin- the guy the Jecklin Disk is named for, revised the dimensions of his baffle slightly and the suggested mic spacing majorly, all because spacing matters.
I did a series of experiments up at Live Oak 10 years ago, trying all kinds of different mic spacings while using the same baffle, from the same spot in the amphitheater. Each spacing sounded considerably different. As a way of reducing the very wide spacing I wanted to use for the omnis, I at one point I considered making two tiny baffles, one for each mic, and placing each relatively close to it's own baffle, then spacing the baffles apart on a mic bar.
Spacing matters with a baffle. Spacing always matters.
I've another suggestion for something to try afterwards on the computer, which can help somewhat when an insufficient amount of spacing and/or angle has been used.