Last Critical Tweak… how to quell occasional high harshness…?

Too many possible culprits. I’ll start with the simplest and the cheapest options-

1. Speaker placement - how far apart are your speakers (center to center); what’s the distance between center of horn driver to side wall and front wall (that’s the wall behind the speakers); This can easily influence the issue you describe 

2. Any acoustic treatments on any of these walls

3. Cables - I’ve had this issue with several interconnect cables I tried in my system   Onece I never had this issues with were Acoustic Zen Matrix Reference MkII and Absolute Copper. These cables are slightly warmer sounding with even top to bottom balance. On a cheap to test what interconnects can do, try these cables from Amazon - get the OCC version, replace all Deuland https://a.co/d/6a8DMZE

Also try their power cord on amp and other components including streamer - https://a.co/d/6nQEACK  

May be even give their OCC speaker cables a try

4. And this is a more controversial topic but this is absolutely critical in streaming - Ethernet cables, routers, other network components. You don’t mention your network config. If you have routers or switches in close proximity to your components and signal cables move all of it away to minimize RFI and EMI. For a more or less decent Ethernet cable try Try this cable - https://a.co/d/0gT7wnr

 +1 @soix. The speaker cables stand out as a very likely culprit. 

I agree  your cable are the culprit.

 audphile1 nailed it. Especially #4.

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Symptoms of un-clean power - link here.

It could be most anything but if it is only occasional high frequency issues you might want to try something like a Loki 4 band equalizer between your source and integrated amp. It has a bypass switch you can use to defeat it when not needed.

But, since it just happened with a system change, if you are up to it financially you might try using different tubes (don't ignore new production tubes, they can work) or new cable (depending on the length needed). I never use less that 12 gauge, most often 10 ga. If you don't understand the differences in cables you might just want to listen to the opposite poles of possibilities on the cheap to see what kind of sound you might like. For example some 10ga Belden and 12 Ga Mogami. Both cheap but will give you some idea which direction you need to be going. 

Logically since you have just 'leveled your system' (whatever that might mean) you might just want to revisit what you are trying to do. If it means you are trying to get a flat frequency response at your listening position you may have introduced the problem which is not consonant with your hearing ability. An imbalance of the frequency response of your ears and system might just be the cause of the problem. For example, the loss of hearing in the highs (mostly overtones) can affect what you hear (emphasized) in the lower range and actually make it sound bright. Not an uncommon experience for some older men who don't recognize the possible effect of age induced hearing loss. 

A complex issue to solve, I don't envy you. :-)