Another vote for Manzanita Lake CG at Lassen. You should be able to easily make it for a couple of days on your battery. Make sure the wiring for your tow vehicle is set up to charge the trailer battery when you're on the road. You should be able to get a full recharge on the drive from Lassen to the Redwoods.
For camping near Eureka, have you considered
Humboldt Redwoods SP? At Burlington CG (which is right on the Avenue of the Giants) you'll camp right in the big trees and you're walking distance to the visitors center. A couple of shower houses there also.
When we want to camp up near Crescent City, we head a little further north over the border in OR and camp at
Harris Beach SP. Nice SP with views of the ocean and easy "commute" distance to the Redwoods. Quite a bit cheaper (something like $35 in CA w/o hookups vs. $28 in OR with electric).