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#JavaScript (ES6), 106 107 113 126 141

A porting to javascript of the Pip answer by @DLosc. I needed some time to fully understand it, and it's genius.

Edit -15 bytes following the hint by @Titus, directly appending chars to the input string a and avoiding early return (so no for/if)

Edit 2 enumerating the 6 value for the Min function saves other 13 bytes

Edit 3 changed c function again. I thought the verbose length and split would be too lengthy. I was wrong.

Assuming lowercase input

a=>[...a].some(z=>c(a+=z)>b,c=a=>Math.min(...[...'lcowbe'].map((c,i)=>~-a.split(c).length>>!i)),b=c(a))&&a

Less golfed

a=>{
  c=a=>{ // cowbell functions - count cowbells
    k = [... 'lcowbe'].map((c,i) => 
          (a.split(c).length - 1) // count occurrences of c in a
           / (!i + 1) // divide by 2 if first in list ('l')
    );
    return Math.min(...k);
  };
  b = c(a); // starting number of cowbells
  [...a].some(z => ( // iterate for all chars of a until true
    a += z,
    c(a) > b // exit when I have more cowbells
  ));
  return a;
}

Test

f=
a=>[...a].some(z=>c(a+=z)>b,c=a=>Math.min(...[...'lcowbe'].map((c,i)=>~-a.split(c).length>>!i)),b=c(a))&&a

;["christopher walken begs for more cowbell!"
,"the quick brown fox jumps over the lazy dog"
,"cowbell"
,"cowbell cowbell cowbell"
,"cowbell cowbell cowbel"
,"bcelow"
,"abcdefghijklmnopqrstuvwxyz"
,"cccowwwwbbeeeeelllll"
,"be well, programming puzzles & code golf"
,"lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. wow!"
,`c-c-b-c
 
i have a cow, i have a bell.
uh! bell-cow!
i have a cow, i have a cowbell.
uh! cowbell-cow!
 
bell-cow, cowbell-cow.
uh! cow-cowbell-bell-cow.
cow-cowbell-bell-cow!
`].forEach(x=>console.log(x+'\n\n'+f(x)))