Instead of starting the push up at millbeck ride into Dodds wood and pick up the footpath at the back of it, that would join up with the millbeck path. It will get you higher up on a bike. There are also a couple of trails in the woods. I couldn’t tell you exactly where the path is in the woods, but it’s at the back ( eastern end) of the wood.
Yes, you can carry up Carlside from either Millbeck or from Dodd Wood.
I’ve done both.
Personally, I thought Carlside direct from Millbeck was an absolute pig (but then it was the second big carry of the day, having just done Blencathra)
If you are insistent on avoiding the scree slope off the top of Skiddaw, then Dodd Wood is probably the better of the two remaining options.