This is a difficult one to answer, mostly because RF and EMI are so incredibly non-intuitive. One might say that if someone claims to understand EMI then they most certainly do not understand EMI. I do not claim to completely understand EMI. I know a lot about it, but I have some holes in my knowledge. Consider that when reading my answer.
My main concern is that LVDS, and really any other differential signaling method that does not use isolation transformers, is not perfectly differential. There are mismatches in the differential drivers that cause common mode "noise" on the diff-pair. This common-mode noise also has a signal return path, which would be on the GND or shield in this scenario. The problem with having the shields disconnected at one end is that this signal return path would be on the power cable-- causing a huge loop area and resultantly huge EMI. While the common mode noise return current is small, the loop area his large, and so this must be accounted for in the design.
In one design of mine, I ran some 2.5 GHz signals over an 18" SATA cable. For those who don't know, a SATA cable has two diff-pairs in it and two shields. Both shields are connected together at the ends. There are no GND wires in the cable other than the shields. In my design, the shields were connected to signal GND at both ends. This design worked great, and is in volume production right now. It complies with FCC Class B, and the equivalent CE version, for electro-magnetic-compliance including radiated emissions, RF susceptibility, and ESD susceptibility.
Going on with the SATA comparison, all SATA motherboard/drives connect the shields at both ends, and they work fine at high speeds. SATA cables are available in length of about 6 inches to 2 feet-- similar to what the OP is using. Systems with SATA meet the more stringent EMC regulations. And they are shipped in the tens to hundreds of millions of units per year.
Were I designing this system, I would connect the shields at both ends. There are millions of modern systems that show this works.
Unless you want to do your own S-Parameter de-embedding mathematics, you must fit a 50\$\Omega\$ connector to at least one end of the trace. You can either fit a connector to the other end, or a good quality 50\$\Omega\$ resistor. I tend to use 2 x 100\$\Omega\$ resistors in parallel for lower ground inductance.
There are many connector styles to choose from, you just haven't looked hard enough yet. If you are only going to 1GHz, then the tolerances will be fairly pedestrian.
If you have a pattern of vias at the end of the trace, you should be able to find a connector with a through hole spill pattern that fits. If not, drill holes adjacent to the signal via through the ground plane to take the grounding spills of such a connector.
If you only want to measure the trace, and not the via, then you have more options. There are many connectors designed to fit a board edge. Cut the via off the board, and fit the connector to the end of the trace at the board edge.
You can solder 50\$\Omega\$ coax to the end of the strip, but you will need to be careful of excess lengths, same pedestrian tolerances but easier to get wrong with cable. Don't tell my boss, but often I would cut a lab 50\$\Omega\$ connectered cable in half, and solder each bit to my test board, saves fitting connectors to cable!
Equipment. A Time Domain Reflectometer (TDR) will give you a nice graphical display of impedance versus distance. A Network Analyser will give you traces of S-Parameters versus frequency, which you would need to analyse to determine the impedances you have. Hint, in the bad old days, a TDR did actually throw a pulse down the track and listened for the reflections. These days a TDR is simply a Network Analyser with an FFT function to synthesise the effect of such a pulse. Both of these types of equipment are very expensive, even to hire for short periods.
There are plenty of ways you can rig cheaper equipment, and some thought, into making measurements of impedance, even if not to 1GHz. A good logic source and a fast digital 'scope will get you a 'poor man's TDR'. A signal generator, a measuring receiver (a 'scope, a power meter, a spectrum analyser), and several tapping points for resistors and a bit more thought will allow good impedance measurements over the frequency range of your source and receiver.
Best Answer
First, note that we're talking impedance, rather than resistance, so you can't just put an ohm meter on it to check.
Differential impedance will depend to some degree on frequency, so it's necessary to measure at the frequency which you expect to encounter, and preferably higher in order to give some safety margin.
The only way to test the cable is to send an LVDS signal through it and monitor the received signal. Make sure the receiver is properly terminated, and use a signal at the same data rate which you will be using.