Although a Vector Network Analyser is the best tool for characterising the Impedance of an Antenna or other RF load,it is not essential-----after all,we don't carry a torque wrench around with us for changing a car's spare wheel.
Often,we can get close enough,using,(as in the spare wheel simile),much simpler tools.
Let's recap:-
The OP wrote in the first posting:- "I've attached 17mm copper wires to the antenna hole"
In the ensuing pages we have pointed out the reasons for the classic forms of antenna construction.
We have all assumed that the antenna will be fed from some form of coaxial feeder,so it is an external device,capable of being tested using a VNA,Scalar Network Analyser or whatever.
My suggestion is that chipwitch produce two groundplane antennas as per a "recipe".
If they work satisfactorily as is,no more needs to be done.
A messy,but possible way to "tweak" the antennas is to trim one at a time for maximum signal strength,using your existing transmitter & receiver.
Here,an SDR would be useful.
Replace the receiver with your SDR,set to give a spectrum display,find your carrier,set it a less than maximum level,& trim your receive antenna,watching the signal strength on the display.
Repeat for your Transmit antenna.
There are other ways,such as an RF sweep,used with a directional coupler,either in the form of a sweeper & calibrated detector,a Spectrum Analyser with a Tracking Generator,a standalone Scalar Network Analyser,or various homebrew SWR meters,return loss bridges,etc.---not to leave out the Grid Dip meter.
Some of the latter start to "run out of grunt" at 400 -odd MHz,though.