Cabinet Cooling

I agree that your best option is to open up the cabinet as much as you can.

Adding fans adds noise.

Basically, regarding fans, you get infinitely more benefit from fans blowing “on” something versus blowing “away” from it. 
As far as dust is concerned; you may find a “Bionaire” or something similar that will provide excellent air filtration as well. Double benefit from the cooling effect :)

https://www.amazon.com/AC-Infinity-AIRPLATE-Thermostat-Cabinets/dp/B00QFWWZQO

I do not know about the noise produced

I have the same problem with my setup and solved it with a small fan clipped to the back of my media unit, directing the air towards the rear of my integrated. I'm assuming the back of your cabinet is open, like mine.

I can hear it between music cuts as it's a cheap fan and it's temporary until I get something like the small, Vornado desk fan, which is pretty much whisper quiet.
 https://www.bestbuy.com/site/vornado-flippi-v6-personal-fan-vintage-white/5064607.p?skuId=5064607

I've tried listening to it and it's so much quieter than the cheap fan I now use. It's just that I'm lazy and haven't gotten around to getting it yet, but with summer on the horizon, I'll be getting one soon. It's easy to position so it's basically a set and forget option. 

As long as the back of the unit is directly in the path of the air from the fan, it'll work like you want. Even my cheap fan leaves my integrated feeling like it was never turned on.

All the best,
Nonoise