Professionally, I did a lot of low current measurements. You HAVE to loose the coax. Connections have to be Triax. Triax cables are about $100+ USD for 3 feet.
The connections consist of ground, guard and a center conductor. Keithley used a graphite impregnated shield to minimize current generation from flexing.
BTW, Triax come in 2 lug and 3 lug. The two lug will fit BNC's but will destroy the Triax connector.
Guard has the same potential as the inner conductor, thus leakage is minimized.
You box should be grounded. Inside your box there should be a separate plate insulated by at least Teflon. I don't know if you have to go better than that. The guard terminal is attached to the second insulated plate.
I'd have to see what I'd have to attach to to make suggestions. One way is to use a metal plate, and a thin shim separated by a Teflon sheet where your device could sit. The metal plate might be sufficient. In the system, I built, there was a grid of tapped holes that we used to attach custom probes and for another case, I designed custom Kelvin probes (other types of measurements) that were initially placed by magnets and held in by clamps using the grid. The design accommodated many 1' x 1" samples.
I really don;t see any reason why silver paint would not work (Used in SEM microscopy) .wires might be wire wrap wire or gold wires used for ultrasonic bonding . In a Hall Effect set-up I designed, the samples were placed on glass slides that had conductive copper tape on them. Wiring was done to the pads prior and Pogo probes could contact the pads along the edge, so the glass acted as a carrier.
A conductive elastomer might work for you, so might a Zebra connector See **broken link removed**. The Zebra connectors are rubber connectors with fine wires in a row and are used too mount LCD displays. They were salvaged from broken equipment.
If you use substrates like glass, they have to be squeaky clean. The rinse I used was semiconductor grade trichlorofloroethane, methanol and Freon TF blown off with dry Nitrogen. Freon TF was phased out because of environmental problems. The Freon TF was a dip. I could measure an appreciable current on a piece of glass with a methanol film on them.
You can also use the coulombs mode, but you generally have to release zero check and hit zero correct shortly after. Divide by the time and you get current.
At these current, both motion (moving a wire in the earth's magnetic field), and flexing (triboelectric effect) create currents and to some extent the piezoelectric effect.
Also look here:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCIQFjAAahUKEwihq86M2PHGAhUMnIAKHWy8D_g&url=https://www.keithley.com/data?asset=6069&ei=VRixVeGsMoy4ggTs-L7ADw&usg=AFQjCNGjLn_QAApj-iiWtbSexwmokhaVVQ&sig2=YORClN1RGcTkDwh0clOESA&bvm=bv.98476267,d.eXY
You might need to use a quartz carrier or better.