Nah, you can get two 600mah rechargeable Li ion PP3 for a tenner. That's nearly 2 hours use swapping one out for the other given the draw you have stated.
https://www.ebay.co.uk/itm/Genuine-EBL-2-4-6-8-pcs-9V-600mAh-Rechargeable-Li-ion-PP3-6F22-Battery-UK-STOCK/153505205743?hash=item23bd9f8def:m:m0X4dtXyta7LE2lpuBjwpIw
Admittedly Im running 6 leds which have a lower current draw, on one such PP3 in my heli and it lasts.
....but he's getting 12 of the regulators, each one of which will do the job, for under a tenner. It looks like modern technology has come up with something genuinely useful. Costs less, does the job, simpler to use that the traditional solutions.
The link shows one PP3 for a tenner, so that becomes £20 for two, but doesn't mention the appropriate charger.* I can't help but feel a bit of uncertainty about Li-ion 9 volt batteries - the chemistry just doesn't allow for actually hitting 9 volts. If it is 3 cells internally, that is over 11 volts, which could well not be good in many uses, in this case offering more volts than wanted thus negating any value of not running directly off the main 12 volt battery. The adjustable regulator also allows the brightness to be adjusted to suit the users preference.
How sensitive are Li-ion batteries to over discharging?
I used to run the nav lights on my small tug (3 LEDS in a series chain with a liniting resistor and bypass resitors to even the current) which drew 15mA, and a non rechargeable PP3 used to last over two seasons sailing.
*there is one on the link under the "related items", £8.99.