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39.95 Eu is ............. 50 USD. We can do much better than that! Try USD 30 !!!(but no temperature facility - we have a separate unit for temp; about 15 USD).(two is better than one!)
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You can buy nice LCD panel display here - analog signal driven. You would need a rectifier op-amp , op-amp in LOG form, and multiplier Op-amp for dB display!

How did you test? Under load, or just open circuit (no load)? Depending on the circuity, it might well drop when under load.

for a good stiffener I recommend the Pizer line of Blue caps ;) ;) ;)

if you really want to drop the 'hum' then use a PI filter - add a choke and your 'stiffening' cap

The second link "http://www.vt52.com/diy/myprojects/other/charger/charger.htm" shows a LM3XX solution and a L200 solution BUT ..........
"The charging current is limited by the small resistor in...

The http://www.talkingelectronics.com/projects/BatteryCharger-12vSLA/BatteryCharger-12vSLA.html circuit seems pretty cool - but you have the risk of 60HZ 'hum' appearing when powered up - perhaps...

"The picoUPS-100 will automatically switch in between V(In) and Batt
depending on who has a higher voltage" from the picoUPS-100 datasheet (pdf file their website)

Is that what you really...

Why on earth use a large cap on a battery PS! A small cap for bypassing IF/RF I can see and would do but 'needing' a large cap for a pure DC source seems ............downright unneccessary.
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"old PSU from a mid-90s computer "

Given sufficient filtering and proper grounding I would not see a problem in using a PC PSU - its just a switching PS with voltages for a PC with standard form...

We are in Shenzhen, China - where a huge number of electronic factories are located and can source cabinets, cases, semi-conductors, LEDs, transformers, caps, test equipment, meters, LCD Panels,...

The Hong Kong companies are usually selling stuff from Chinese mainland factories - they might say that they 'have' a factory but chances are they are acting as distributors.

We are in Shenzhen,...

I think you have 2 challenges; selecting the amp, and incorporating the battries with a recharger.

Given batteries of sufficient power (mAH) i would think it would superior to a regulated power...