We didn't get them at HF, but those are the same lanterns we have. Man, those things are BRIGHT!
Haven't had very good experiences over the years with rechargeable batteries though...I'll just use lithium AAs.
For the rechargeables, the NiMH or Ni–MH work well if charged between ambient and down to 35F.
Often the small sized NiMH are used in solar lighting. These are a variation of the old school Edison battery, but sealed with a unique modern structure.
NiMH batteries typically have two to three times the capacity of NiCd batteries of the same size, with significantly higher energy density, although only about half that of lithium-ion batteries. NiMH batteries have almost entirely replaced NiCd. That stated, the NiMH do last longer over plenty of the cheap, leaky Alkalines people tend to use today. The Lithium-ion will give you almost a full month of runtime, however there are limited chargers for these and the charge temperature window is much narrower of ambient down to 50F. There are trade-offs no matter what is chosen.
The perfect scenario is to choose what battery technology works for you.
I use both NiMH and Lithium-ion due to various applications or cost effectiveness.
Let's say, you need lighting and to charge your cell phone and/or laptop. You own some power tools like Ryobi, Ridgid, Dewalt, Bosch, etc. You actually have stored power right there at your fingertips as a power block cell.
Skill is taking the lead by directly putting the battery as a power block cell for direct use.
Milwaukee has the power block cell adapter,
For your vehicle, the jump-starter boxes are plenty too.
There are several options out there. All in compact packages to meet all sorts of needs.