I'd also recommend you ask for route and timing help on the Backpacker forums as there are a couple guys that live in the area and could offer alternative routes rather than those of us that can just generalize on things.

It's purely conjecture here, but perhaps you could try a different trailhead or reverse direction or plan your route to camp only in the neighboring NF (permits from them should be far easier to obtain) since the crest follows very close to the park boundary. For example, my son and I are hiking into Yosemite from Inyo NF since permits are much easier to get and avoid the hassle of Half Dome permits to boot (at least for this year).

Even in good weather though, I think the TCT is very aggressive for "novice" backpackers. Surely you're not planning the whole 60+ miles in 4 days/3 nights? How much experience do y'all have with mountain hiking?