Does the issuing of a Letter Before Claim fall come under the banner of 'conducting litigation'?
I'd say not. Schedule 2 of the Legal Services Act (2007) defines "conduct of litigation" for the purposes of the Act:
4(1)The “conduct of litigation” means—
(a)the issuing of proceedings before any court in England and Wales,
(b)the commencement, prosecution and defence of such proceedings, and
(c)the performance of any ancillary functions in relation to such proceedings (such as entering appearances to actions).
A Letter of Claim is issued before proceedings are issued/commenced.