I managed to revive a Caster HPS ballast by wiring in a red barbecue piezoelectric ignitor in parallel to the lamp and using the barbecue ignitor to start the lamp manually. It works because the barbecue ignitors work by generating a high voltage pulse to spark the burner on a barbecue. In a lamp circuit it acts as a ignitor. 

I have 4 cobraheads, 3 gumballs, 2 remote ballasted mercury lights, and 4 yardlights.

1967 GE M250R Crime Fighter - 175w MV
1976 Thorn Nite-Liter - 175w MV
1980 Powerlite B2255 - 400w MV (Converted from 250w HPS)
1993 Sylvania R47 - 100w MV    (Converted from 250w HPS)
1996 Cooper UCD - 70w HPS
2011 Cooper OVF - 400w HPS

1950s Large Powerlite Gumball - Mogul Incandescent
1950s Small Powerlite Gumball - Medium Incandescent
1950s Line Material Sphero-Lite Jr. - Medium Incandescent

2x 1960s Revere Urban-Lite - 175w remote MV

I got pretty much all of my lights from local sellers, antique/flea markets and the occasional Restore find. Aside from the Crime Fighter and the Urban-Lites, the rest are representative of what was used in the Toronto area.   

Yeah, they're pretty neat and much better alternative to installing HPS. The majority of the gumballs were installed in the mid 90s and replaced multiple incandescent gumballs. The model used in the initial install are Cooper URB which were discontinued in 2008ish. Newer installations use bare aluminium King Luminaire PSMH teardrop fixtures.

Have a couple of them, both have incandescent lamps in them. CFLs take too long to warm up (especially in winter), they only get used for short periods of time, so the power use isn't an issue.

A 150w HPS GE M250-R2
A 200w HPS AEL 115
2 of those 20w Warm-White LED 'eBay streetlights'

For all you West-coasters (and everyone else) eBay seller "tonhackamac" has 7 line material gumballs for sale. eBay search "Antique Looking Aluminum Stadium Lights" and they should pop right up !

“I’m rolling in my grave!”
Signed,
Tom Edison

Even most of their light bulbs and locomotives are junk now. The newer 2017-2018 fluorescent lamps only lasted 1-2 years. And their locomotives have lots of problems with certain things blowing quickly.

Try to search through the forums here, this topic was discussed through already a few times, including step by step manual (with links to the required graphs) to determine the correct ballasting lamp or combination for a given discharge spec.

First most cameras have 5VDC supply input on some connector, usually asking for 2A rating. So connecting any 5V/2A "mobile charger" to that connector should work.
Otherwise:
What batteries it takes?
If R6 (I guess so from your photo), you may count anything in the 1.1..1.5V per cell, round it to the nearest convenient voltage for an available regulator.
If it is a LiIon cell, 3.3V should be OK (if the converter is adjustable, you may set it to the 3.6 or 3.7V, anything within the 3.2..4.2V range should be fine)
Industry standard voltages are 2.5V, 3.3V, 5V, many convertors are actually adjustable.
Then use some DCDC step down converter module (E-bay, AliExpress, Banggood,... are full of these), rated for at least 2A output and set or settable for the voltage you need for the camera (see above). Place this module into the battery compartment and connect instead of the battery, with input wires going through the battery cover.
Then supply this from some 12V or so of suitable power rating (count the module efficiency as about 60%).

For 5V you may use directly some 5V brick, then you need an low ESR capacitor (in the 1000..2200uF range, rated about 10V) placed in the battery compartment in the camera to eliminate the inductance of the cable.
Using the converter has an advantage in the possibility to supply its input directly from the car 12V socket...