This turned out much longer than I thought it would. Skip to "conclusion" at the end for a tl;dr.
I would like to build a custom DIY battery pack that uses a larger battery.
This is a good idea!
Can I connect the battery directly to the flash?
This is not!
Sadly, the links are dead and I can barely find any information on the product. What I did find seems to be that it uses 117V on the light while accepting both 12V DC and 120-240V AC as power input. That leads me to think that it has a voltage regulator built-in, but that doesn't have to be the case. I will assume it does not and instead uses the 12V from the battery pack to power it's circuits and then converts this to 117V for the light. This is much safer to assume and usually how it's done.
As a comparison you can picture a smartphone powerpack. The batteries inside run under 3.7V but the circuit (voltage regulator) inside the powerpack steps this up to 5V which is then provided to your smartphone as steady as possible. Since the battery pack you're talking about also has a charge indicator, I'm sure it has a voltage regulator too.
What type of batteries will you use?
If you use a car battery it will typically be rated 12V, but that doesn't mean it's always going to give you exactly that. An almost-empty car battery outputs around 11.6V (can go as low as ~10.5V, which is VERY bad for your battery though). Fully charged it will give you around 12.7V. While this whole range should be perfectly fine for your studio light, it can be very bad for the battery if no protection circuit is added. The studio light could easily discharge it beyond 11V which can permanently damage the battery. Bad idea.
If you use multiple cells things get even worse. You could chain three 18650 batteries together and get 12V out of that, but there's two issues: The first one is the same as above. All empty those batteries will give you around 9V (dangerously low for the batteries), all charged around 12.9V. The real issue here though is that the cells might not be charged/discharged balanced. I've had plenty new 18650 batteries from the same vendor before with completely different resistance.
So how do I make it safe?
What you want is a circuit to give you two things:
A) Regulate your voltage so you get a steady 12V out of the batteries
B) Protect your batteries from over- and undercharging
What you need is a battery protection board.
This will ensure that your batteries are discharged and charged evenly and won't over- or undercharge. What you need to take care of here is the maximum ampere rating. In this case it's 50A.
Luckily, unlike with the studio light, I was able to find a manual for the Bowens Travel Pak. It says in there that it usually charged to 500W, 750W flashes are listed in there aswell too, so I assume it accepts that too. 750W would mean (750/12) 62.5A. The board I linked above supports a maximum draw of 50 Ampere, so that might not be enough. What I've found about the studio lamp however says 500W, which would be 41.67A, so it would work for that. If you want to be on the safe side, get a board that's rated higher than 50A.
This was only relevant if you're using multiple-cell batteries!
Now in addition to that you also need a voltage regulator / boost converter. You will also need one if you want to use a car battery. Again, you need to make sure it can provide enough power (500W / 42A @ 12V) for your light. This is kind of a problem if your input (the battery) is 12V (unregulated), since voltage regulators usually need some room to work with. It's easy to sind 24V to 12V and vice-versa, but 12V to 12V I'm unable to find anything on eBay and amazon that will give you at least 500W of power. You can find out here why.
There's two solutions to that issue:
A) Get a 24V battery (or batterypack). Use this to step it down. Cost intensive though.
B) Connect the battery directly. If you use multiple cells with a board I've linked above this could be a significant problem, as the board will allow the cells to discharge to around 3.05V. Which, times three, would give the light just 9V to work with. If you use a car battery this shouldn't be too bad, but you have to keep in mind that undercharging the battery can permanently damage it. I wasn't able to find a discharge protection board for 12V batteries that can handle more than 500W.
The Bowens Travel Pak manuel also says this about its battery connector:
Gives a direct connection to the internal battery, not current limited or regulated.
This leads me to assume that the regular output does regulate the voltage to 12V.
There's a lot more than just voltage that you need to care about. If your studio light depends on regulated 12V, connecting a battery directly will not work. If you want multiple cells (e.g. 18650 batteries), you need a protection circuit aswell. Your light needs 500W and I was not able to find a voltage regulator that can do this if your input from the battery is 12V aswell. If you are able to find one (I really don't know any sources, just looked at eBay and amazon) then you can put that in-between the battery and the light and you will be good to go. The manual of the Bowen's Travel Pak suggests that it regulates it's output voltage, so your DIY battery pack should aswell. If using a car battery directly, you should really get a discharge protection board, I wasn't able to find one that can handle 500W, but they should exist.