After all this time I'm pretty sure DSP is gay, but I think he might not even realize it. So it's not like he's closeted and that Kat is his patchy beard, I think he's just following the signals that society gives him and his parent's expectations and going through the motions of happily married man.
He probably doesn't have sex with Kat but rationalizes it somehow—we're busy, we're adults, our schedules are exhausting. All his attempts to demonstrate his heterosexuality were weird and conspicuous; that boob painting and the ostentatious Black Cat bust. He'll watch wrestling and play WWE Champions, staring at muscle-bound men wearing tights and grabbing each other, but if a woman shows up in a game, even a hard 4, he fucking averts his eyes like he's a kid going shopping with his mom in the underwear section.
I think he'd be a lot happier if he practiced some self-examination about this but he's never done that about anything so I'm not holding my fucking breath.