5 mm Red LEDs are like a 2V zener with an ESR of about 10 Ohms. White are 3.1 V and 12 Ohms ESR
What size are these around the lens? 5?8?10?(mm)
8 LEDs of this type in parallel is still like the same Zener voltage but now ESR/8 Ohms or between 1 & 2 Ohms ESR. (Zener specs use Zzt to mean ESR which is about 1/Pd the max power rating of the low voltage diode.)
Did you actually measure the battery voltage Voc, Vload and use Ohm's Law to compute the result. This will tell you the exact divider ratio of the battery ESR and Load ESR. Since AAA batteries have an ESR of about 1 Ohm and AA about 2/3 of that give or take depending on Voc / pulse short circuit current=ESR.
Ohm's Law will tell me exactly what Voltage and source impedance to use, or current rating.
You never use a CC source because if one LED fails, the rest increase their share of current.
I might try a 1 Ohm series R from a high current low voltage supply like 3.3 V or a couple Ohms with 5V ATX PSU. Then the voltage drop in the known R value tells you the exact current being shared.
Come on guys, this is basic Ohm's Law. The only trick is understand the ESR or 1/load regulation delta v/delta I of the source and the load. The ideal match is to match the load ESR to the source ESR such that the source voltage drop matches the load voltage rise . e.g. RED LED rises from 1.8 to 2.0 over 20mA approx or 0.2V/0.02A= 10 Ohms. while White may rise from 2.85 to 3.1 V or 250mV/20mA = 12 Ohms. Normally a 25% tolerance and good ones will be lower ESR, and poor ones higher. Also higher power LEDs will be lower again where ESR=1/Pd power rating of LED.
If in series, then add the Vf and ESR's to match the equivalent load.
I've sold about a million of these types (previous 10 yrs during my retirement) of 5mm LEDs in RYGBwhite types. with very little variation. I still have 20k excess if anyone wants some , send postage or buy 200 for any reasonable offer, and I will mail it to you. The LEDs in each bag are the brightest in the world and all ESD protected.. e.g. 16000 mcd 30 deg ANSI neutral white or 20000 mcd or 10,000 mcd RED or 12,000 mcd yellow 30 deg. The angle affects peak brightness with lens ~2x ((1.8))for each reduction in beam-width /2.
Each batch is so perfectly matched they can all be run in parallel without risk of thermal runaway ( <<1% matched Vf)
People use CC regulators in large power LEDs so they can substitute a wide range of suppliers with >25% tolerance and deliver a constant shared current in series or use a resistor drop of at least 3 V in strip LEDs so it can be operated from a car from 11 to 14 .2V at varying brightness.
These LEDs in your sign will be fairly well matched if from the same batch, so a CV with supply ESR to match the load is ideal. If each LED is 8mm is 100mW and they are RED then you need a 400mA at 2.2V maybe , guessing size and quality of your sign LEDs, more if poor quality, but if one battery was weak, or the LEDs are poor quality, you may need more. If they were clever, they would have hidden a series R equal to the tolerance of the ESR for those LEDs in each bulb. (to equalize or reduce the variation of If on each chip, the shunt current. Or perhaps they added 10 Ohms in series raising the Net ESR . again Ohms Law rules above threshold voltage, Vth where (Vf-Vth)/If~ESR using about 10% of rated current for Vth