Determine the "Luck" of a string

JavaScript (ES7), 123 112 107 bytes

s=>2**[5,4,3,2,1,0].find((i,_,a)=>a.some(j=>s.includes("luckyL".substr(j,i))))-2*~-s.split(/[omen]/).length

Edit: Saved 11 bytes thanks to @Titus by assuming that the letter L does not appear in the input. Saved 5 bytes thanks to @Oriol. ES6 version for 125 114 109 bytes:

f=
s=>(1<<[5,4,3,2,1,0].find((i,_,a)=>a.some(j=>s.includes("luckyL".substr(j,i)))))-2*~-s.split(/[omen]/).length
;

<input oninput=o.textContent=f(this.value)><pre id=o></pre>

05AB1E, 36 32 28 26 bytes

Œv'¸éyåiyˆ}}¯é¤go¹'ƒÖ¦Ãg·-

Explanation

Œv         }                  # for each substring of input
  '¸éyåi  }                   # if substring is part of "lucky"
        yˆ                    # add it to global array
            ¯é¤               # get the longest such substring
               go             # raise 2 to its length
                 ¹'ƒÖ¦Ã       # remove all chars from input that isn't in "omen"
                       g·     # get length and multiply by 2
                         -    # subtract
                              # implicitly display

Try it online

Saved 2 bytes thanks to Adnan

Ruby, 91 87 bytes

String#count's finnicky usage strikes again! (When passed a String, it counts all occurrences of each letter in the function argument instead of all occurrences of the entire string.)

Try it online

->s{2**(z=0..5).max_by{|j|z.map{|i|s[b="lucky"[i,j]]?b.size: 0}.max}-2*s.count("omen")}

A version that takes in lines from STDIN and prints them: 89 bytes (86 +3 from the -n flag)

p 2**(z=0..5).max_by{|j|z.map{|i|$_[b="lucky"[i,j]]?b.size: 0}.max}-2*$_.count("omen")