Hi OP,
Doppler distortion is a real thing, but usually the room interferes more. It's worth understanding if your bass is smooth enough before going much further.
The quick answer is you can put a high pass filter in front of your amp, which is just a single capacitor at the right value for the amp impedance.
I think 250 Hz is a little too high to cross a main speaker at. Around 100 Hz is usually the upper bound, 60 Hz the lowest I'd normally recommend, but the idea that adding a subwoofer AND utilizing a high pass filter on your mains is sound, especially if you have room modes which you want to use an EQ to tame. This way you put the EQ in line with the sub only.
Also, check out the AM Acoustics room mode simulator which will help you understand how bad your room modes are as well as give you ideas for the best placement.