Supposing climate change continues at its current "worst case" projected rates, would we see a noticeable reduction in insects down the line? I am wondering if it makes sense to have a character who has noticed a change like this in their lifetime, or if such a change would be unlikely or not drastic enough to attract the average person's attention.
Yes and No.
Even today it is easily possible for older people to notice the change. When you for example compare the amount of insects on your car window from 40 years ago to today it is easily noticeable that the insects have decreased.
So why no?
Because this decrease is not caused by climate change (at least for the most part), but due to air pollution, with for insects toxic gases. These do not need to be green house gases.
Scenarios, which are according to IPCC are based on ultra pessimistic assumptions but technically within bounds of science, would give you global temperature higher by 5 C at the end of century. So within remembered lifetime you may get maybe 3 C.
While sudden temperature shift would cause (speed up) some extinction, insect have short life cycles and are ultra adaptable. Migration of insects from warmer climate zones is also likely. Maybe, if your character is really good, he can see how species changed.
Actually, I think that a scientifically accurate post apocalyptic settings in which temperature is a few Celsius higher and sea level raise haven't reached one meter has a great potential. Those scenarios assume high (but dirty) economic growth. So you may end up with main character who is quite affluent person by contemporary standards and who complains about electricity bill caused by combination of heat wave and excessive use of air conditioning.
It would depend on how drastic the damage done by climate change was. Also, are you only including climate change in this anthropogenic destruction? Sea acidification and anthropogenic species extinction are both affected by and would affect insect biodiversity too. how far into the destruction are we talking? How far in the future, and how much more damage? I think for a noticeable drop off in insect species there would need to be atmospheric level changes that would also change vegetation, and that would probably throw a few more bones into your future system. So, plausible? Maybe. But it would also probably be an extremely uninhabitable future. Think no insects means no naturally grown/pollinated crops, too, unless you have some system of artificial pollination going on. I would also the question the term "post climate" since any climate is a climate. A true no-climate world would probably be more like the moon since there would be no measurable indicators. But that's just semantics. Cool question.
Insect population as a whole won't be reduced by climate change. What will, and does, change are the species which you encounter.
We have already problems with asian hornets coming to central europe, and there's the fear that mosquitoes (and with them malaria) will spread, as they will be more and more able to survive the milder winters.
In turn, local species have trouble adapting and fighting the "invaders", so they are reduced; but the total number of insects will probably even increase (disregarding other pollution effects).