That sounds a bit like my Blasa! She's more shy with her coughing and doesn't tend to "let it rip" like the others do. I've always described their cough as sounding similar to scraping my fingernail across the edge of cardboard (I get bored and "test" sounds of things...

) - it makes a very dry raspy sound. Their oxygen levels seem well maintained during
occasional coughing - no one turns blue or seems to be struggling.
A chest x-ray should have honestly been the first thing done when you brought him in for coughing! The longer this goes on the more likely the heart will be affected even if the heart is not the original source. My vet charged around $80 for the x-ray. A specialist is not needed to do this unless the regular vet doesn't do x-rays.
I kind of hate to suggest this in case it is his heart, but maybe get the x-ray ASAP and then use that to decide if more expensive tests are really needed? I sort of wonder if you weren't pushed into more expensive testing too soon.
Have all infectious causes been eliminated? If not, depending on how the lungs look, a tracheal wash can show if there's anything in the lungs or if it's just inflammatory.
How is Hobbes energy level? Does he seem to be playing a bit less or or breathing hard during play?
I have one who's more cough resistant, but I can see it in her stamina during play - she tires and has to take breaks just like a person would. The male doesn't really seem to do this unless he's really bad off... but then he's a young boy cat and I haven't had a young male cat yet that doesn't try to push boundaries a bit.

This is why he's most likely to have a coughing fit. The third one, Blasa, doesn't really do much of anything ever and never has - so I have no stamina gauge on her.
Now here's the problem for you: All cat litters are bad. Clay, grain, wood, silica, dusty newspaper (though lesser for this). They all make dust or have allergens in them. I have a cat that has problems with clay, hence the reason I haven't used it in nearly 4 years. I wasn't using it when the kittens came along. I was using chicken feed and grains are common allergens in animals and people. Whether this triggered it or not, I will never know. Swheat is wheat, and could be problematic because of that.
I now use DIY versions of the smart cat box with safflower seeds. One person in another thread mentioned using popcorn kernels

in much the same way. A store bought option is Tidy Cat Breeze. In all cases there is no odor control (just keep the boxes clean) and urine drains into a bottom box - some use pads to absorb it, my DIY boxes just collect it which I then flush. (less litter going to a landfill!) You have to wash the boxes often, though.
These can be harder for cats to adjust to because they're not soft litters, though I've not had problems. Due to the life span of the litter replacement material (I don't know what else you'd call it?) and the amount you use, cost is comparable or a little less than litter is.
To start with, I'd eliminate everything else you possibly can in the home as it's not difficult to stop using scented products, dust/vacuum more often, and change air filters more frequently. If it turns out Hobbes lungs aren't the source of this, there's no harm in cleaning.

ETA: Mine started coughing at ~10 months old.