I think for your setting, a directional antenna would help sometimes and not others - I assume the home bases for the cars are all in the same area, so most of the time you'd be able to keep them out of the antenna's focus, but there would be times when you couldn't avoid pointing at them, and you'd probably have to fish around for the car you were looking for sometimes - could be a lot more work.
A bandpass filter will keep high level stations on other bands the receiver can pickup out, but it doesn't sound like they're contributing significantly to your overload. Maybe next time you're out, take a broader look at the spectrum before the race and see if there's anything else strong getting close - if there are, then maybe the bandpass filter would help.
You could try bandstop/notch filters to remove specific strong stations, but if they change frequency or if there are several, it could be a real pain to try and chase down. At least theoretically, you could have an adjustable notch filter that you have inline with your antenna where you look at the spectrogram of the bandwidth coming in, and then tune up or down to pick out the strongest signal, you could largely remove it from being a problem.... but if there's more than one, you start needing more notch filters and the whole thing gets complicated fast.
You still have the attenuator-on-the-input option if you can differentiate the weak signals in the noise. It sounds like the airspy mini may do the trick for that, getting a couple extra bits worth of usable data beyond the more standard 8 bit converters, so if you haven't already tried it, bring a modest attenuator to check inline with that - maybe the new hardware will give you that low signal performance. If you have adjustable input stage gain, you can also just turn it down, which spares you from needing a physical attenuator.