How to proceed?

@stuartk ...my previous system (same speakers + dac, with Wells Audio Majestic integrated and Simaudio transport) was not at all fatiguing and there was not a trace of sibilance.

...One dac that interests me is Border Patrol, which is very much designed with particular attention to power implementation.

This is a key indicator to me. Since the new integrated and new transport was introduced, the issues you describe started to occur. Also, re-check your path of any new "silver" based cables (power, interconnect) you introduced around the same time. If you have stock power cables lying around, try all together at once.

btw, I own the new Border Patrol SE-I dac, while I agree its smooth and non fatiguing, if I may suggest, I don’t think the dac change alone is going to resolve all of your noted issues. I kind of suspect its a combination of things going on now. Imo, as a test, I’d borrow/swap the amp and transport and [all] cables back out with stock ones all at once. Then go in reverse putting new pieces back in play one-by-one to see if it surfaces again. Worth trying to help understand if you can narrow down a specific culprit (or combination of things) from there. Best of Luck.

@decooney

Thanks for your suggestions. They make sense. Unfortunately, I do not have any unused gear lying around and don’t know anyone who would loan me the gear.

One exception-- I tried my old tubed Joida CDP as a transport and the sibilance was worse.

@stuartk my understanding is you have tried different cables between dac and amp and that didn’t solve the issue. Those were DH Labs cables. 
In my experience DH Labs cables are brutally honest and lean slightly on a brighter side of neutral. 
I will give you an example from my most recent cable swap/upgrade experience…

I was using Acoustic Zen Absolute Copper XLR between the Bricasti M3 DAC and XP22 pre, as well as between the XP22 and X260.8 monos. 
The AZ Absolute Copper were a bit too warn for my system with the latest upgrades. So I called The Cable Co and tried the following: Audience AU24SX, Analysis Plus Silver Apex, Kimber KS1126. I had on hand AZ Silver Ref II and the Nordost Tyr 1 and Tyr 2 that my friend loaned me. 
There was a significant increase in sibilance with the AP Silver Apex. A bit less with Audience AU24SX but still present, there with AZ Silver Reg but no sibilance with Kimber and Nordost Tyr 2. I never experienced sibilance in this system with the AZ Absolute Coppers. There was always a super smooth midrange with delicate treble but the bass was too plump with my Wilsons and the Pass monos. I struck gold with Nordost Tyr 2 XLRs. My speaker cables are Audience AU24SX 

So the moral of the story is - you still have not done due diligence with the cables. 

@audphile1 

Thanks for your suggestions.

I have used The Cable Co. in the past and felt very positive about them but my last experience was so negative, I'm not sure I'd want to go back. 

I had A. Zen Silver Reference and they worked great in my previous system but were too bright once I acquired the Hegel and Jay's. So I tried the Absolute Copper but they were extremely dark sounding, top to bottom. 

I'd be very happy if my problem could be solved via cables.

AZ Absolute Copper are warm. But don’t forget with higher resolution components like your Jays cdt and Aqua DAC + H390 that may be a welcome change. I ruled your speakers out - there’s nothing there that will give you that sibilance.
If you can, try the Nordost Tyr 2. Used they’re not terrible price wise for the sound upgrade they bring to the table. You can sell them if they don’t work out but I doubt that will be the case. They’re excellent cables. Much more refined top to bottom than the older Nordost lines which I wouldn’t recommend as a remedy for brightness and sibilance.