What is purpose of this exercise? What do you try to accomplish?
Scope BW is measured by using leveled sinewave generator. Directly to 50 Ω input on scope. On scope input that is high impedance (1MΩ), an 50 Ω passthrough terminator is used to keep signal generator properly loaded. Same when measuring with a probe as a system.
For a risetime measurement you need pulse source that have (very!) flat top and pulse should be at least 10-20x wider than risetime. Same type of termination.
Using super fast edges won't help, it will only incite more resonances in probes... it is more important for an edge to be clean, linear and monotonic...
It only has to be reasonably fast. Risetimes of source and scope add as RMS so it is easy to calculate scope risetime if using known source... As an experiment you can try and use a much longer coax used as a storage transmission line on your pulser to make pulse wider.
There were several topics here where VERY detailed info, explanations etc was shared about theory and practice of passive probes. And it was shown that (nominaly) low BW probes aren't necessarily low BW. If you have low enough source impedance..
Predominant influence to measurement by passive probes is their load capacitance. That is, they won't act as lowpass filter internally (so much) and not pass through high frequencies as much as they will "bog down" (load) signal source (the basically form an RC filter from source impedance and its shunting capacitance) and shunt signal to the ground... That is why your two probes that have same load capacitance will have similar performance...
Passive probes BW spec is more of a "use it up to this frequency" recommendation. In practice with high impedance sources, passive probes are not very "dependable to show the right thing" above some 150 MHz anyways because of the loading.... Despite saying 500 MHz on a probe..
Higher BW probes also will usually have two additional compensation pots..
Procedures:
https://www.euramet.org/Media/docs/Publications/calguides/EURAMET_cg-7__v_1.0_Calibration_of_Oscilloscopes.pdf summary:
1. Modern scopes in general will have brickwall AA input filters and will not have perfect pulse response even with source directly connected to BNC on front.
2. In order to have comparable results for discussion a standard procedure has to be followed: in your case 50Ω source parallel terminated with 50Ω terminator to scope input and probe input. Only one at a time.
3. BW cannot be determined with pulse source (risetime measurements) you need sinewave generator with 50 Ω source impedance and precise levels (or a way to independently measure calibrate level).
4. Using pulse source, a BW can be roughly estimated (roughly) but pulse source has to have certain characteristics (and you need to know your scope AA filter coefficient. It won't be 0.35 for your scope. And it will change with how many channels are on). Pulse source is used to calibrate levels and pulse response. Clean edges and flat top and pulse width at least order of magnitude wider than risetimes.
5. All of that is good and nice if search of knowledge. In practice, if you need to look at 150MHz + you need to start to think in terms that your probing is part of circuit.. A 10x passive 50 Ω probe made from 450 Ω resistor and a piece of coax will do better job and load circuit much less at 200 Mhz than a 10pf 500 MHz 10MΩ passive probe...
6. Which brings us to the point that in that case you should have bought a scope that has 50Ω inputs if these kinds of signals and measurement is what you need. I personally would not buy a 200MHZ+ (less than 1.5 ns risteime) scope without 50Ω inputs...