I've done the stretch of the A6 from Shap down to Kendal several times before and that's been fine. It's a fast road, but on the way up the hill it's a double lane, so plenty of room for folk overtaking you. Once you're over the top it's a fun road to cycle - twisty in places, but you can get a fair speed up and I didn't suffer any dodgy overtakes (although I'd avoid it around rush hour).
I think I'd be less keen to cycle on the A6 between Penrith and Shap - It's a faster road and just doesn't have the feel of a road I'd want to cycle.
A more pleasant (but hillier) road from Penrith to Shap is via Askham and Bampton Grange, which has the advantage of nicer countryside, Shap Abbey and the fact it passes a number of locations from Withnail & I. The road via Ullswater and Kirkstone Pass is considerably hillier and would take you much more out of your way.
Leaving Kendal, I'd go via Natland (the A6 becomes dual carriageway, so no fun on a bike).
Further south, all anecdotal, but Milnthorpe seems to be a cyclist accident blackspot. I don't know the specific circumstances, but it's probably a busy road at rush hour.