It's pretty common for high bandwidth scopes to use some fancy BNC, though they may be proprietary, it shouldn't be too difficult to source something since they're used in pretty much any scope with BNCs and >1GHz bandwidth.
For just measuring something, not replacing the connectors, I'd just make sure you attach either a BNC cable rated for >2GHz to the connector or a BNC to SMA (or other) adapter rated similarly. Cables/connectors rated for the same or more bandwidth that are from a reputable manufacturer should be able to do their end of it, and as they are BNCs, they should be able to mate properly with well made BNC connectors.
Honestly, though, your measurements may be in spec. Total uncertainty on a signal generator can easily be 1.5dB over a few GHz span on a calibrated instrument (though probably not typical), then a little loss from the cable on the higher end and the normal rolloff on the high range still puts you above -3dB at the stated bandwidth. Do you have a spectrum analyzer that you can compare the response of the cable and generator with? Or a power meter?
Better yet, that video you linked says you can de-embed the characteristics of your cables/adapters if they're measured on a VNA, which would give you the best picture of the actual frequency response of the frontend.
In any case, I've done some similar FFT measurements of swept signal generator sources on high bandwidth scopes with regular BNC cables and have seen similar level artifacts. Remember that you're getting somewhat less than 8 bits of dynamic range over the full scale (as zoomed in as possible will maximize your use of the available dynamic range) with an accuracy spec of 2%. If you zoom in a bit and compare it to an SA with similar settings you probably will be pretty close.