Include a shielded cable with the sensor. Put a sticker on the sensor saying: shielded cable is required.
Spend enough time to understand why your ADC readings are sensitive to the missing shield connection. Maybe you can improve this by changing the filtering on the signals on the inner conductors?
For a product like this I have seen good results from connecting the board ground to the enclosure (not sure if you have done this, but I see no reason not to).
Also I have seen this type of product with a full metal enclosure and only one shielded cable connecting to the outside being very immune to ESD and burst tests (CE) as well as very silent on EMI. So unless there is something I haven't picked up, you should worry if that is not the case.
Electrolytic capacitors do have a mechanism whereby some DC measurements show different values. Once you are in the audio band, it's likely to be constant from 20Hz to 20kHz.
The question is, what method does your Fluke use to measure their capacitance? Does it apply 1kHz AC, in which case the measurement will be true for the audio band as well. Or does it apply a DC current, and measure how quickly the voltage changes over a period of seconds? This latter method is prone to errors from leakage current (big anyway with electrolytics, especially horrible with new ones), and from charge absorption (have you ever watched the voltage reading on a big electrolytic that's been charged and quickly discharged grow again, over the course of minutes?)
As the DC leakage depends on the charge history of the capacitor, there's no reason that similar DC measurements should correlate to similar AC measurements. So it's important to find out what measurement your meter is making.
Why 10% and 20% tolerance caps? The purpose in life for an electrolytic is storing energy in power supplies, not much else. They don't need a good tolerance, so they're not built to a good tolerance. The form of construction doesn't lend itself to good reproducible dimensions anyway. But with automated assembly on high quality machines, it's not surprising to find several capacitors made in the same batch to match to within percent.
Just leave generous clearances and you should be fine. Even with well designed and made boxes (eg. Hammond) it's not unusual to allow a couple mm overall clearance (1mm all around). Eg. (from above datasheet- maximum recommended PCB size)
Maybe you want to allow 1.5mm rather than 1mm if it looks a bit rough. Shrinkage (typically a couple of percent in linear dimensions, first-order compensated for in the mold design) in injection-molded parts is affected by resin choice and processing parameters so it's more likely to vary if the manufacturer is swapping resin types after the mold is designed or is pushing for high production rates.
Keep in mind that there are always draft angles in molded parts (not always shown on drawings) so the inside will be smaller at the bottom (top of the mold core) than at the top. Otherwise the part would not come out of the mold easily (or at all).
Similarly, allow generous hole sizes for mounting holes- use at least the "loose fit" diameter in mechanical engineering sources\$^1\$. As well as linear shrinkage, bosses can bend a bit (the part is a bit soft coming out of the mold) if it is not handled perfectly.
\$^1\$You can find that information online- look for "tap drill" tables, but the real bible, in North America anyway, is Machinery's Handbook.