DISTORTION IDEAS NEEDED

The Tech 21 Steve Harris pedal. I don't do guitar, not sure it'll do what you want, but it's a LOT more neutral than the dUg and has more versatility, including the ability to alternate between 2 different saturation levels.

Bump for additional suggestions.

 

100% subjective.

You can get great growl and OD on a bass from many preamps or OD pedals. But know some dirt pedals don’t work as well on bass as guitar for any number of reasons. So there’s nothing wrong with trying a few to find your sound.

For light OD I love a BB style like from Browne or Wampler or straight from a tube preamp pedal, Sushi Box has some great ones with clips online. The Wampler Pantheon has Baxandall bass EQ which is a feature I’d look for unless you get a bass-specific product. Spaceman makes several good OD options.
For a more pure distortion there are a lot of Rat types that work well with bass.
My personal flavor if I’m dropping dirt on bass is fuzz. Maybe it’s just me. But I think the wooly fuzz effect works better for lower frequencies. I have close to 40 fuzz pedals and I can say some of them are almost completely transparent on bass even though they sound great with guitar. So make sure you get something that is Bass friendly.
My subjective favorite bass fuzz is the Asgard 3 by Blackhawk (Baxandall EQ) But also check out Wren & Cuff custom shop because they let you select Bass specific mods.
No wrong answer. Have fun.
Pedals are like potato chips. You can’t have just one.

 

What you want is an amp in a box style overdrive/distortion in particular one that has a Mesa/Boogie gain and saturation profile and tone stack. Trying to emulate the complexity of a Mesa/Boogie amp in any pedal is a very tall order for any pedal designer. There are a lot of takes out there but I will offer two suggestions.
Joyo California Sound is a solid state version originally designed by Andrew Barta of Tech 21 for the Character series but discontinued. This pedal can get a wide range of Mesa tones utilizing the Character control and is one i have used for many years in y studio on bass and guitar parts with satisfying results.
Sushi Box Wrecking Ball X offers a tube version of the Dual Rec lead channel in a pedal that I plan to add to my collection as soon as funds allow.

 

For power amp squishiness, maybe look at a BJFe Blueberry or derivative (like a Blackberry or a Honey Bee). Stay away from the Mad Professor Blueberry, though, as that one sounds completely different (more or less a fuzz).

 

Plug all nine stompboxes you listed in series, turn them all on, and then lower your standards re: how closely that sound mimics the Mesa 5:50.