Or the more frequent hourly bus to Scarborough (especially as the HF house at Larpool is on that route- the X93/X94, so you would have to go into Whitby first to get the fairly infrequent Whitby to York bus). The X93/X94 bus goes to Scarborough station. Then train to York and on to Bristol.
The X93/X94 to Scarborough is the route the HF suggest, and the one I would too.
https://bustimes.org/services/x93-middlesbrough-scarborough-northstead-2?date=2024-04-21&service=X93&service=X94
1 hour on the bus to Scarborough, 50 minutes on the train to York every hour, then 3 hours 45 minutes to Bristol every hour.
By the way I hope you have enjoyed the good weather at Keswick this weekend. I was actually in Keswick this afternoon, for longer than planned due to a broken down bus.
I don't see any need to cancel the North York Moors. This is an easier journey than appears at first glance. The trick is not to take the train out of Whitby to Middlesborough (only a few a day), then Middlesborough to York, direct every hour- purely due to the infrequency of service and the slow route to Middlesborough.