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Single AA thrower will be blown out of the water by single 18650. Convoy C8 in warm should do the trick, they don’t do high CRI option.
Thrower = flashlight with big reflector 
I love my Jaxman M8 with the Nichia 219b. It’s bigger than a AA light, but smaller than a C8.
I don’t love the modes, but since I do use it occasionally on my bike it’s nice to have strobe.
ETA: And you can get it warm (4000K) and high CRI.
Convoy has the S6 and another S (S3 maybe) with a deeper reflector than the S2+. Ask Simon on AliExpress to assemble one with the tint you want, biscotti driver (has mode groups with and without strobes) and smooth reflector. won’t throw as much as a C8, but I don’t think anything will for the same price.
The Emisar D1 fills all but the price requirements if you are willing to go up to $40, tried and approved by many forum members
EDIT: I figured since you asked for AA that small size was desirable, but since you mentioned already have the C8, it’s just a matter of ordering an emitter and driver that sills your needs
Klarus Mi7 is relatively throwy given its size. Works well with a 14500!
I would suggest the Emisar D1s ( https://intl-outdoor.com/emisar-d1s-thrower-p-926.html ) or the D1 if you want something smaller.
Astrolux S3
Good throw, bright with 14500 cell, reasonably priced.
Manker T01
Emisar D4s - It is available with Luminus SST-20 3000K or 4000K 95 CRI
Convoy C8 (Simon’s AE store) - It is available with Nichia 219C 4000K 90 CRI
If you can spend just a bit more over 20$US, you can get the Night watch NS22 from this thread for about 23$US:
With the 4000k/3000k SST-20.
Bought one of these in 4000K , liked it so much that I just ordered another in 3000K with the 21700 tube with both.
Manker T01 II has a ton of throw and is available in neutral white, although I don’t know the CRI. It’s definitely not cool white, though. And as a bonus, unlike half the flashlights people have suggested in the thread so far, it actually runs on AA/14500 like you requested. The cost is a little higher than you’re hoping to spend, but it’s an actual thrower, not just a mini flashlight with some throw.
On a budget, the Thrunite T10 II is one of those inexpensive, mini flashlights that just happens to have a tight hot spot. It throws just a little better than something like a Lumintop Tool AA, and the neutral white version is very warm indeed, downright orangish. I’d estimate the tint is around 3000-3500 Kelvin. Again, not sure about the CRI, and it’s not a dedicated thrower, but the cost and tint meet your criteria.
Just noticed that. I’m sure the individual has found what they were looking for by now.
Thanks for the heads-up on this light. Free shipping too. And a discount coupon. I ordered one with the SST-20 4000 95CRI and 21700 tube.