As a Keswick local, I’d suggest Ambleside, Esthwaite (Hawkshead) or Coniston. There are a greater density of legally rideable trails. While people post how good Keswick is on here, a lot of those trails are footpaths, and are quite committing – especially in mixed weather. Look at Postie Rich’s Jenn ride route – 100 miles of brilliant, legal riding around Ambleside.
If you are heading to Keswick, stick to the Borrowdale Bash, Tour of Skiddaw and legal riding out around Buttermere and Loweswater.
Just my opinion, as a local rider, bearing in mind it’s peak holiday season.
EDIT – of the YHAs you list, Borrowdale. Helvellyn is the wrong side of a big mountain (though great riding around Ullswater), Ennerdale is the middle of nowhere and Buttermere, while beautiful, is also very rural.