Hey Mike!
Welcome! I lurked a LONG time before posting, so I know exactly where you're coming from.
I did the HTC a couple of years ago (#66) and I still chase other exotic trout all over the west, so I also know the motivation. 31 salminoids and counting...
As far as Warner Lakes Redbands, I think those waters will be just fine, since the elevation is so high, but the flows might be really low, so that might be an issue more than temps. Goose Lake RB? Not sure, if you do fish for them, I'd suggest the higher the elevation, the better. McCloud RB water might be pretty warm too. I got one of those in June when the flows were a lot higher and the water was still ice cold.
Lahontan Cutts, same as the Warners, a high elevation stream (at least where I caught them), so temps should be fine. Eagle Lakes RB, I failed to catch in a stream, catching scores of small brookies instead, so I can't honestly say on that one. I didn't try the lake at all.
I think the Kern River RB and Goldens will be fine, the goldens being found in a high mountain stream, and the Kern RB are in big (and cold) water, so I think you'll be fine.
If you're going to try and cram all of those places into one week, I'm sure you're aware that's a LOT of miles on the highway and on the trail.
Either way, I'd suggest you post a report of your trip when you get back, for the rest of us to live through vicariously.

Have a great time!
Cheers!
--FlyGuy (Dave)