How to proceed?

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. 

How old is your Aqua? Chances are it won’t change too much but most times I’ve found DACs do change after 400-500 hours run time (though in my case they usually got a bit more detailed). But if it’s new you may want to run it constantly for a few days just to see if anything changes. If I recall correctly I think in one review of the La Voce the reviewer found it a bit strident on the top end compared to some other DACs. I know you said you didn’t want brand recommendations but you may want to look at Mojo audio DACs based on what you said about the Hegel vs the Aqua DAC. They have nice organic tone but are not rough on top at all, and mine sounded best when run from a good transport.

@jimmy2615 

Aqua is well broken in by now. 

Thanks for the Mojo suggestion. As it happens, I was all set to replace the  Aqua with a Mojo but Benjamin expressed concern about my transport, so I bought the Jay's instead (couldn't afford both). 

@jimmy2615 ...If I recall correctly I think in one review of the La Voce the reviewer found it a bit strident on the top end compared to some other DACs. 

I vaguely recalled a similar comment and looked back a few years to try and locate when folks were comparing the S2 vs. S3 versions of the La Voce dacs, and found members on another forum changing out caps in the S2 in an attempt to tame some of that with Jupiter Beez Wax caps and such. Not sure how it turned out. Maybe something there worth following up on to learn more.   

@stuartk 

I checked out your system page to get an idea of your room- beautiful house! Your options for speaker placement don’t seem to be working in your favor though.

The open doorway behind your left speaker and the right speaker almost in a corner are enough to cause issue. Not sure how WAF plays into your situation but have you considered rotating your monitors 90 clockwise? 
I doubt the sibilance your experiencing is a by product of your dac (assuming it’s in perfect working order). I believe the La Voce (while producing a very good presentation) is the warmest of the Aqua line when it comes to high frequency reproduction. I have the La Scala Optologic MKII and I’ve never noticed or felt the presentation to be sibilant (outside of what might be within a particular recording). I also believe purchasing a Border Patrol would be a step backwards in your case. 
I cannot comment on how the dac within your Hegel plays into this but I would definitely wager that it is not of the level of quality of your La Voce. While I cannot speculate as to what it’s doing differently in this case my guess would be that its presentation is just more rolled off and less transparent than that of the La Voce (but again that’s a guess). Any higher quality dac you purchase will probably present the same issues but with more magnitude.

Just as an experiment have you tried placing light acoustic materials around the various surfaces in your listening room just see what effects it may have?