The Becherovka did not play as well with the other elements as I would have liked.

The flavor profile is good, but quite boozy, which adds heat while somewhat masking the herbal notes of the amaro and aquavit unfortunately. (Note that Frederic Yarm's version uses Rittenhouse and Linie, which is how I made it, coming in at 85+ proof undiluted.) I like the use of Linie here, but for a second try would opt for a lower proof rye, perhaps basic Overholt at 80 proof, to see if that creates a better balance of flavor and alcohol heat. I am rating it at 4+/5 without making an adjustment to the rye choice.

A bar spoon of Demerara could also round things out for you. I've gotten used to NYC drinks that use even less sweetener (sometimes 1/2 oz amaro or even 1/4 oz falernum total) and if it's still hot and not soothed by ice melt before things fall apart, I use some syrup (Dem for anything with brown spirits or dark amaro, and simple (or agave) for white spirits) the next time to give a better balance to the proof/bitter-sweetener ratio. Or sometimes in the glass to patch things up.

I selected Rittenhouse since I find its spice notes complement most aquavits well, and better than most other ryes.

The drink is sweet enough, I wouldn't want to add any more sugar and unbalance it. The problem with running too hot is that at a high enough level it can mask some of the flavors--a reason for adding just enough water through dilution of one sort or another to release some of the flavor notes. This can be a tricky balance because sometimes the heat is contributing useful flavors as well.

I did a test last night cutting 1/4 oz of the Rittenhouse and diluting with 1/4 oz additional water to get it in the target range for mixing. Unfortunately, this did not reveal much more herbal character in the finished cocktail. The sweetness of the drink was more apparent this way (despite not adding any additional syrup/sugar), but the rest of the profile was about the same as before. My conclusion is that I was getting most of the flavor/aroma already with the original recipe. I thought there might be more waiting to be uncorked, but that was illusory.

'A strong drink' - no, ya think?!? I used Bounty (Fiji) in place of the Hamilton, so no idea how different that would be. It's good, but if you build it, give it a little dilution time.