Forget any linear voltage regulator IC. You wont be able to manage the losses, plus it will eat up a lot from your expensive battery energy.
That circuit seems to be very simple, but it is by far not trivial, mainly when aiming for some usable efficiency. Plus there are quite a few design errors on the primary circuit, mainly yielding poor reliability.
First the transistor choices: The 2N3055 are really obsolete junk. Extremely inefficient for a switching circuit. First they are very slow, then the 0.1 Ohm saturation voltage slope is not that great either and even to reach that you need 10% base current. With all that you would be wasting more than 10W in transistors and the base resistors, plus have to design the circuit with really low operating frequency (the 20kHz range mentioned in the article is way above what these transistors could work in reasonably). Better would be something along the lines of MJF15030 for 3- or 4-cell (10.8 or 14.4V) battery supply design, or 2SC6144 for 2-cell (7.2V) design, both allow base current to be as low as 1/20 of the collector current.
Second the base feedback winding uses way to high voltage - with the winding ratio and the 12V supply, it exposes the emitter base to reverse voltages in way excess of 16V (assuming clean rectangular voltages without any overshoots), while they are rated at barely 7V. With 30turns CT primary, the feedback should have about 4 Turns CT, not more. Plus a series resistor of 1Ohm or so should be added to each base, to limit the reverse breakdown current during overshoots.
For low operating frequency these transistors imply, the size and cost of the secondary ballasting capacitor would be rather high (you have to look at its AC voltage rating at the given frequency, co 150VAC often implies more than 1kVDC rated capacitors)
Better would be to switch to MOSFETS (e.g.
this topology). The transistor voltage rating should be 5..8x the maximum battery supply voltage, Ron in the 30mOhm range, with the modern "logic drive" FETs (they use to feature lower gate charge, so the gate bias resistors could be higher so dissipating less) the gate Zener should be about 6V. Plus side is, you dont need the feedback winding...
The drain inductor could be anything in the 10..30uH range, a choke from PC supply would work well...
And don't forget the core gap is, what controls the frequency (along with the ballasting C7 capacitance - the transformer secondary forms a resonant LC with C7), so it needs to be carefully adjusted.
