How are you with going into a store like Muji? (They showcase their home fragrances at most if not all their stores)
And non chain Chinese and Thai restaurants? (Many of these will keep personal shrines)
It’s easy enough to avoid incense at the big temples - even if you buy a packet of single use stick on face masks, avoiding temples with smoke is very straightforward, so you shouldn’t need many.
But if your sensitivity to irritants is peculiarly high, it’s worth being aware that incense is extremely prevalent in ordinary people’s lives. A great many Japanese people (and I even include myself when I’m there) start every day burning incense. So it will be on many of our clothes and hair.
Small restaurants and businesses may well have their nook where incense is burned.
I stress that this isn’t a scent most of us notice or are aware of, but if your intolerance is extreme, you may experience sensitivity to this hidden and very subtle side of Japan.

I suggest buying a package of the dust masks that construction workers use before you leave home. They look enough like the kinds of masks that Japanese people wear when they have colds or hay fever that people will probably just assume that you're wearing some weird kind of foreign allergy mask.

Unless the OP is affected with some kind of synesthesia where odours trigger discomfort/nausea/migraines.

I admit that I don’t really understand what LP needs to avoid, but he may not know either.

Occasionally, walking through perfume halls in department stores, I get sneezing fits and/or painful, watering eyes. And yet I may well walk the same route through the same department on another day and am entirely unaffected. Goodness know what the trigger is.

I avoid fragrance sections of stores.

Never had a problem.

Alone, staying away isn't a problem. My wife would like to go, though.

At that level it's not an issue.

Yeah, identifying the triggers can be a royal pain.

No reason why she can’t, you aren’t tethered together.

Check to see if Denboin Temple will allow access to their lovely gardens on the dates you are visiting. If so, that would be an ideal place for you to hang around whilst she’s at Sensoji in Asakusa.

Japanese incense is often made from food-grade powdered organic (as opposed to inorganic) ingredients and as such often doesn't trigger scent-related reactions. I found this out when I was first introduced to a line of Japanese incense after years of wanting to vomit around bamboo-dipped stick incense sold in head shops, kitschy boutiques, and on street corners. You might find it really doesn't bother you that much.

As for wearing a face mask, it's typical in Japan. No one with question you about it. You'll fit right in around the bigger cities.