A lead-acid battery powering a Joule-Thief??Don't use rechargable batteries for a Joule thief, once discharged to this level they can't be recharged.
Rechargeable lithium batteries also do not like being discharged too low.
But the rechargeable Ni-Cad cell in each of my solar garden lights gets almost completely discharged every night for years without any problem.
The voltage stepup circuit is similar to the Joule Thief and a few of them do have the Joule-Thief circuit.