How do you pronounce the surname Beauchamp? Do you say "boo champ"?

Bee-Cham - one of a few classic counterintuitive pronunciations, 
like Marjoriebanks  Cholmondley Featherstonehaugh 
(Marshbanks Chumley Fanshaw)

In the UK we  pronounce it "BEE- cham".

In the UK the English pronunciation given to it is beecham.

Bow-champ. (like a bow and arrow)

It depends on the location.  In the northern US, beech-um, in the south, bo-shahm, with a long o, and shirt a. 

Depends on French vs. English pronunciation.

Bow-shom or Bow-sham.  The p is usually silent.

The first half sounds like bow (as in a bow and arrow) and the second half sounds roughly like shamp.  But the 'a' in the second word is similar to an  'o' but not quite the same.  I hope that helps - a little.

Bow-Champ, Bow-Kamp is also another way to pronounce it depending on dialect...

N.Shadows

it is pronounceed dossunbaribari

See https://forvo.com/search/Beauchamp/ for the French and English pronunciation.