Although product recommendation requests usually get closed on this site I have a couple of comments to make.
First off you should be aware that you pictured DeWalt's 20V Flex Battery technology. Yet you showed a DeWalt charger that is designed for a different 18V setup. You need to be very careful to not mix up technologies. Use the correct manufacturers ...
9-volt batteries are made up of six relatively small 1.5V cells which do not deliver much current. As you may know, 9V batteries are often used in small devices that are meant to run for a long time (smoke detector, for example) but draw very little current.
The Energizer 9V (522) battery datasheet shows the battery's milliamp-hour capacity in the following ...
In addition to Marcus Müller's suggestions, consider that battery packs often end up as rectilinear prisms. So take different stacking arrangements of finite numbers of cells (3, 4, 5, 6), stack them "square" and stack them "triangular", then calculate the area of the smallest enclosing rectangle.
Somewhere on the Internet there will be a guide, but for ...
Assuming two A23 batteries in parallel at \$12 V\$, the charge available is about \$110mAh\$ ( \$55 mAh\$ each). So, converting the \$12 V\$ to \$5 V\$ with average efficiency, the capacity will be equivalent to \$220 mAh\$ at \$5 V\$. So, it would last about 220/330 hours or about 40 minutes.
The ideal way to transfer charge between two batteries with substantial voltage differences is with a buck converter, with efficiency able to be in excess of 90%.
However, at low voltage differentials the use of a resistor may be more efficient!
Power input = Vin x I. Power Output = Vout x I.
Efficiency = Pout/Pin = Vout/Vin.
For eg 12.5V in and 12V out, ...
The temperature rise actually helps generate more power because the chemical diffusion and reactions are faster at higher temperatures (between 20 degrees and 30 degrees C the capacity goes up by 10%). The loss of capacity is due to reaction products clogging up the surfaces, side reactions, and limited diffusion of reactants.
Speculation- packing volumetric density is not that important, mass is by far the important factor. Secondly, the need to have heat drawn out (especially during charging), typically through liquid coolant actively pumped through the battery bank.
Congratulations on doing it properly as you can in your situation, on adding an extension in a competent manner and in thinking about other ways to achieve your requirement.
Fortunately, it is likely that you have done something wrong with your extension cable. I say "fortunately" because it shoul be possible to make what you have done work.
Perhaps the ...
Your LED is clearly operating as a switching regulator. The current draw is almost certainly not steady, unlike an incandescent lamp. How do you know your current meter was reacting properly? If you used a true RMS voltmeter to measure the voltage across a known small series resistance, and calculated RMS current thereby, how would it compare to your ...
A simple workaround is to measure voltage under load and without load. So you can calculate the change in internal resistance. Regarding the capacity of the battery, it is important to know under what discharge conditions this capacity is calculated. Very often this is a 20 hour discharge with low current - conditions too favorable for good results.