First of all, rechargeable NiMH batteries have a more consistent voltage than alkaline batteries. Even your current headlight might be more consistent with a set of those batteries, and it will certainly be cheaper in the long run than constantly feeding it alkalines.
If gradually dimming output is a problem, then you ultimately want a headlamp that is regulated for consistent brightness, or what Petzl calls "constant lighting." They don't give clear info that I'm finding on which of their lights have a regulated mode, and I don't think their brightest modes are regulated, but I think the Tikka and Actik are. I'm not sure if Black Diamond models are.
From brands you typically can only find online, like Fenix, Nitecore, Armytek, and Zebralight, there are numerous very good regulated models. Some run on AAA batteries, some run on 1 or 2 AA batteries, some run on the more expensive CR123A lithium batteries. There's also a few out now that run on built-in lithium-ion batteries, like the Nitecore NU30, which is probably one of the highest performance lights of a similar size to your Energizer. The NU30 battery is USB rechargeable, which can be very convenient, although it will eventually wear out over time (hopefully not for at least 5 years), and is not replaceable.
Even more performance can be had from headlamps that use the larger 18650 lithium-ion batteries, but these almost always have a top strap.