Maybe an over night stop in Ellenboro is needed.
08:25 train from Leeds, via Shipley, then change to a bus at Skipton, arrives Carlisle 11:52. Bit of a ball-ache journey as there is engineering work at the weekend:
From Settle-Carlisle website "Trains will start from/terminate at Skipton with connecting buses operating between Skipton and Shipley to transfer to trains to/from Leeds"
Bus from Carlisle - service 301 departs 12:01 - (a bit tight as its 6 mins walk from rail station) arrives Whitehaven Duke Street 13:55.
If thats a bit tight then there is an earlier train from Leeds 06:45 arriving 11:12
Return journey :
Bus departs Whitehaven Lowther Street 18:10 service 301
Arrives Carlisle 19:58
Train from Carlisle :
20:07 (again a bit tight) arrives Leeds 23:23 1 stop via Preston
Or 21:15 from Carlisle arrives Leeds 00:51 1 stop via Manc Picadilly
A long day but can be done