Never measured temperature but even after 10 plus continuous hours of playing neither amp has ever become too hot to comfortably hold hands against sides for as long as I’d care to do so.
If thermal expansion/contraction properties of different metals were the only explanation, why wouldn’t every BHK300 exhibit this phenomenon?
Rates are different…
I would suppose that different BHKs would reach different maximum temperatures.
Room temperature, ventilation, and how hard they are driven would all be a factor.
My BHK 300s heat sinks reached a maximum of 112 degrees F last night and no clicking noise.
If yours do not get significantly hotter than that, then I wonder what the difference is?
Any chance you can get a temperature measurement of your heat sinks?
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No, I have no way of measuring, however, to give you a less specific idea, they never get too hot to hold hands against each side of either amp for as long as I’d care to.
So, either I would have to turn them on one and a half hours before I start my listening session… or don’t use them at all. Most noisy equipment I ever owned were the active Manger Studio Monitors (MSS). They were humming and hissing quite a bit. I was fully aware of that before I decided to invest those 14k Euros. Because they sounded just incredible (still the best I’ve ever heard). I never really got used to these noises, but the SQ was worth it at that time. At least it was a constant background noise. But intermittent metallic clicking? No way! That would drive me crazy. If I remember correctly, the amps for the bending wave transducers were rated 1kW (seems to be a lot, not sure, if that is true, but Manger wanted the amps to be extremely fast), they were heating the room, the amps for the bass were not as powerful but did get very hot too. But they never ever clicked. I do miss those speakers… o well 
I don’t get any clicking from my BHK300s now. One of them was making audible noises due to a faulty input board. After getting it fixed there are no unexpected sounds (either from the speakers or the amps directly).
You could ask the same question of a lot of other products in the range. There are owners here of two DS DAC’s, or two Power Plants, or two of whatever else, where each of the pair are not identical twins.
Thanks for posting. I have the same issue with BHK 300’s clicking when warming up and down, at time intervals similar to what you describe.
I respectfully disagree with @Paul and others (Corvette and 911 owners) downplaying the issue. The amp’s purpose is to amplify electrical signal in order to play music. Any noise that is not music is distracting. A click that occurs reliably is akin to a noisy person sitting next to you at a concert. You can accept it if you can, or walk away. I don’t see any other options.
This cannot be dismissed as a non-issue because I heard it, and I was forced to investigate. Because you know, audiophiles are obsessive. I had to stop the playback and rewind and listen again to see if the click is on the recording. I had to do a number of isolation tests to identify the source of the problem. I had to search for the answer online and found this discussion. I read it in full, hoping to see a satisfactory resolution. I am typing this comment now. All together, it took hours that I would rather spend doing something else. Multiply that by the number of BHK 300 owners in the world, and you probably get the number of hours large enough to affect the GDP of a small nation.
As @anon85907017 said, it is indeed interesting what people are willing to accept from their expensive gear.
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By the way, instead of tightening the screws, I suggest loosening the screws. Rather than trying to counteract the forces caused by different rates of thermal expansion, prevent the build-up of tension. Let the different parts contract and expand at their own rate. The assumption being, the resulting displacements are small enough that the whole assembly can still hold together. Loosening the screws slightly will allow the displacement to happen gradually, rather than in steps. This should hopefully help with the issue.
It is not a non-issue, but endemic to many amplifiers and potentially annoying. However, once one knows the cause it is fairly easy to accept it.
I like the idea of playing with the screw tension on the heat sinks . Just do not loosen them so much they can no longer do their job. 
Please let us know if you find something which works.
A quick calculation with thermal expansion numbers. Aluminum expands about 8 times more than steel: 3 microns per 1 degree Celsius per meter for steel vs. 23 (same units) for aluminum. So, the expansion of steel is dwarfed by that of aluminum, and we can ignore the expansion of steel. The height of the BHK 300 heat sink is about 20 cm. Assuming the temperature increase of 50 Celsius, we get linear expansion: 23 [microns / (m * C)] * 50 [C] * 0.2 [m] = 230 microns. So, about a quarter-millimeter. Not a huge linear displacement, probably of the order of the manufacturing tolerances. And definitely not enough to worry about the chassis falling apart.
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I have have had this “ping” issue on 300’s for a long time and though I can
rationalize the science behind heat, contact, and dissimilar coefficients
of expansion, the consistent sound is annoying. I have had many brut amps,
even class A ones from Mark Levinson that dwarf the 300’s in size and
heat, but no ping. Can’t say it’s the nature of the beast if only this one growls.
My Classe CA-400 had the same chassis pings on warm up and cool down.
Understood, and I remember some amps doing this back in the day, but
mine continue to do it during play, not just warm-up and cool-down. A bit
disconcerting during a quiet passage.
Many years ago I had a 200 lb behemoth of an amp. A Krell KSA250. Every now and then I got sound that resembled a tuning fork pinging. Krell told me it wasn’t a big deal just the heat sinks doing their job.
Maybe some metallic components are just a bit too tight.
I did lessen the pinging of my CA-400 by taking the top and bottom covers off and re-installing the zillion or so screws just snug instead of finger stinging tight when I broke them loose.
Thanks for your kind input. This seems to be the consensus on a fix.
One would think that in the assembly of heatsink to chassis, the proper
torque would be assigned to the fasteners.
I have no idea what the construction method is but if it is simply screws then you could put a hard rubber (or maybe felt) washer on the screws between the metal components. That would allow movement and tend to cancel the stick-slip noise.
Playing with the screw torque may or may not work. Try it and let us know.