Here is my take: gigabit Ethernet is 125Mbaud, around 80MHz bandwidth, my guess of edge rate of a couple ns, edge travel distance of 1/3 of a meter. That should be a lot longer than the travel distance of the transition through the connector system. So the exact impedance does not matter that much.
Of course you want to avoid connectors with excessive capacitance and inductance. I think one way to check that is to measure the travel distance of the signal from the termination point at the PCB to the point where the cable unraveled. If that is shorter than the compliant RJ45 set up, then it is probably no worse than the RJ45 arrangement.
This also points out that when terminating the cable to the mating connector, it is necessary to control how much the cable get unraveled (un-twisted).
By the way, in the old days, PCB terminated RJ45 jacks almost always have the spring contacts on top (locking tap at the bottom). So the wire of a contact has to travel to the back of the jack and then down to the PCB. Now most jacks are flipped "upside down" so the traveling path becomes significantly shorter.