Yes Geoff, there is a Santa Claus.
We've run lithium batteries to exhaustion without noticable problems with the InReTech AA adapter. Craig has expressed general concerns about thermal issues, but neither of us have seen the magic smoke leave the adapter in our tests. Keep in mind that neither of us has the funding and nobody has the time necessary to establish a good baseline for how much damage, if any, might done by running our adapter on lithiums exclusively. We did take into account thermal concerns when designing the InReTech AA adapter.
You can run lithium batteries with the InReTech adapter and it makes the light brighter for quite a while longer. It might do damage to the adapter over the long haul. It might only give you six years of continuous use instead of twelve. If you can afford to keep feeding your light lithium batteries for six years of continuous use, you can afford a new adapter. I strongly suspect that the economics of the matter are that after 200 sets of lithium batteries, you probably will want to change the adapter and that adds 1% to the price of your batteries.
I would say that the performance really does depend on what you are using the adapter for. If you are using the light continuously, the lithiums simply give a brighter light longer. If you want to give this light to your grandchildren, you might choose to stick with alkalines. For some applications, lithium batteries might be too bright for comfort. I think that alkalines might be superior for camping because it is easier on night vision if the center beam is not blindingly intense.
I have not tested the InReTech adapter on the Brinkman. I will look for one and try to get you some feedback on that.