I don't think to wrap antenna or ever ground it will change situation.
Unit will continue transmitting data, whatever noise it can pickup,
let say it's own input circuitry.
Protocol analyzer, sniffers and other technological "breaking enigma" option is
good, but only for federals with unlimited budgets.
I'd suggest, it'd be easier to decode message that is not a noise, but
something repeatable over and over again.
if it's power monitor, get one more same kind device, that will transmit in close proximity to your monitor, stable power signal over the time.
1). Turn on transmitter, set power that definitely will suppress any environmental garbage.
2). Make a dump of the stream from receiver, or watch it on you serial monitor attentively.
3). Change a power of your own RF source.
4). Take a second dump, (or don't look around if it stream on serial monitor -):
5). Compare results.
a) If no difference, most likely, bit-rate settings mismatch (receiver and analyzer).
Change serial settings of the analyzer, go to step 1.
b) if you notice the difference at least in one byte, it could be data-byte RSSI.
Go to step 3, play around with Tx signal several times, to gather more information.
Hope it will help.