Yes, a passive crossover will work for this purpose. Carvin sells one that mounts right into a cab, if you're willing to drill a hole into one or the other of your cabs that might be a way to go. If you get a passive crossover, make sure it will handle the full power of your amp.
But, you can do something much simpler than that. Just connect a large non-polar capacitor in series between the 2x10 and whatever it's connected to. That will keep the lows out of your 2x10. It'll have to be a fairly large capacitor, and it'll probably have to handle a decent amount of voltage, depending on your amp's power output.
To calculate the (approximate) value of capacitor that you'll need, use the following series of formulas. These are "DC approximations", but they're close enough for all practical purposes.
First, you'll need to know the impedance of your 2x10 cab. I'm guessing it's 4 ohms. Then, use the formula t=RC to determine the value of the capacitor you'll need. The 't' is "time", which is the inverse of frequency, so if you do all the math you'll end up with C= 1/(f*R), where f is the frequency at which you want to start limiting the lows, and R is the speaker impedance.
Here's a f'r instance: Let's say you have a 4 ohm cab, and you want to set it up so that nothing below 200 Hz gets into it. C=1/fR, so 1/(4*200) = .00125, and this will be the capacitance in "farads", which equates to 1250 microFarads. This is a fairly large value. It'll probably be impossible to find a non-polar cap that big. So, what you do is, put two 2500 uF electrolytic capacitors in series, back-to-back (in other words, the two negative terminals are tied together), and that will give you what you need.
To figure out the voltage that will be required, use the formula P=(E*E)/R (or "E squared over R"), where E is the voltage. Doing the math again, E=sqrt(P*R). Use the "peak-to-peak" power to calculate the required voltage (usually this will be about triple the amp's rated RMS power). Again, here's a f'r instance: if your amp is rated at 100 watts RMS, and you're using a 4 ohm speaker, the RMS voltage will be sqrt(100*4), or about 20 volts. The peak-to-peak voltage will be about triple that, or about 60 volts. To be perfectly safe, try to get a 100 volt capacitor. That will give you plenty of headroom.
Note that this method will not keep the highs out of the 12" cab, but will keep the lows out of the 2x10. It sounds like this will accomplish the desired result.
In practice, and in the above example, the cutoff at 200 Hz isn't "instantaneous", instead there will be a "smooth rolloff" that starts slightly higher in frequency, and the rolloff will get more significant as the frequency decreases. So for instance, at 200 Hz you'll allready be at about 1/3 of your signal strength. At 50 Hz you'll be down to around a few percent. At 20 Hz you'll be down so far that you won't hear a thing. This will protect your 2x10 speakers. If you wish, you can experiment with the value of the capacitors to get the rolloff characteristics that sounds best to your ear.