Firstly, the pedals that I like the sound of with bass are Big Muff, Lovetone Big Cheese and (strangely!) the old Boss Distortion yellow pedal. I've got round the bottom end problem like this...
My amp has an active and a passive input. I put the bass into input A of and A-B box, with output A going to the passive input, and output B going to a small chain of pedals going to the active input. The pedals I used are the Big Cheese, eq pedal and a Lovetone Doppelganger (obviously you don't need this for distortion!). You can set the eq to compensate for the distortion pedal and use the A-B box to kick them both in at once, using the output on the eq to match the levels. If your amp doesn't have two inputs, there are pedals out there which just provide a true bypass loop, so you can switch in a whole chain at once. There is this one
TheGigRig Loopy 2
This is quite expensive but I'm sure you can find cheaper equivalents.