I rode this a couple of years ago but in reverse. My route is below. I turned inland at Duddon Bridge and guess this is the route being suggested to you. The A595 was fine all the way to Duddon Bridge, very little traffic, and I only turned off because I had been warned the A595 was a poor road for cycling. You'll be able to see just north of Ulpha I hit a climb of 13-17% on a loaded tourer - and I travel light!!!! Once up this climb, I walked, there is then a long 9/10 mile descent to Eskdale and beyond to Holmrook. Apart from the vicious climb it's a lovely ride. The problem you face is you will have to climb the ten miles from Holmrook to Woodend. It's long, long climb.
Personally I'd take my chances with the A595, the section I rode didn't have enough traffic to cause me any concern. The Bay Way is fantastic.
https://ridewithgps.com/routes/17446560
Halfway between Eskdale Green and Holmrook I met a lady in her 60s on a sit up lady's step through with loaded rear panniers. She'd ridden over from Ambleside via Hardknot - I felt shamed on my ten speed triple