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s => // s = input string
s.replace( // replace in s ...
 /[a-z]/gi, c => // ... each letter c (case insensitive)
 ( //
 s + // if the previous letter was a vowel
 // (or this is the first iteration)
 g( // or the current letter
 i = // whose ASCII code - 1 is loaded in i
 B(c)[0] - 1 // 
 ) ? // is a vowel:
 "" // append nothing
 : // else:
 "ei" // append "ei"
 ) + ( //
 h = j => // h is a recursive function looking for the
 // replacement letter
 g(i) ^ // if the type of the current letter
 g( // does not match the type of
 j = -~j % 26 // the next letter in the alphabet
 // obtained by incrementing j modulo 26
 ) ? // then:
 h(j) // keep advancing by doing a recursive call
 : // else:
 B([ // output the letter
 j + 1 | // whose ASCII code is j + 1
 i & 96 // with bits #5 and #6 taken from i
 ]) //
 )(i & 31), // initial call to h with j = i mod 32
 B = Buffer, // define B for Buffer
 g = i => // g is a helper function taking an integer
 s = 1065233 // representing an ASCII code minus 1
 >> i & 1 // and returning 0 for consonant / 1 for vowel
) // end of replace()

s => s.replace( // replace in the input string s ...
 /[a-z]/gi, c => // ... each letter c (case insensitive)
 ( //
 s + // if the previous letter was a vowel (or this
 // is the 1st iteration and s is still a string)
 g( // or the current letter
 i = // whose ASCII code - 1 is loaded in i
 B(c)[0] - 1 // 
 ) ? // is a vowel:
 "" // append nothing
 : // else:
 "ei" // append "ei"
 ) + ( //
 h = j => // h is a recursive function looking for the
 // replacement letter
 g(i) ^ // if the type of the current letter
 g( // does not match the type of
 j = -~j % 26 // the next letter in the alphabet
 // obtained by incrementing j modulo 26
 ) ? // then:
 h(j) // keep advancing by doing a recursive call
 : // else:
 B([ // output the letter
 j + 1 | // whose ASCII code is j + 1
 i & 96 // with bits #5 and #6 taken from i
 ]) //
 )(i & 31), // initial call to h with j = i mod 32
 B = Buffer, // define B for Buffer
 g = i => // g is a helper function taking an integer
 s = 1065233 // representing an ASCII code minus 1,
 >> i & 1 // returning 0 for consonant / 1 for vowel,
 // and also saving the result in s
) // end of replace()

Try it online!

s => // s = input string
s.replace( // replace in s
 /[a-z]/gi, c => // each letter c (case insensitive)
 ( //
 s + // if the previous letter was a vowel
 // (or this is the first iteration)
 g( // or the current letter
 i = // whose ASCII code - 1 is
 B(c)[0] - 1 // loaded in i
 ) ? // is a vowel:
 "" // append nothing
 : // else:
 "ei" // append "ei"
 ) + ( //
 h = j => // h is a recursive function
 g(i) ^ // if the type of the current letter
 g( // is not equal to the type of
 j = -~j % 26 // the next letter in the alphabet (wrapping)
 ) ? // then:
 h(j) // do a recursive call
 : // else:
 B([ // output the letter
 j + 1 | // whose ASCII code is j + 1
 i & 96 // with bits #5 and #6 taken from i
 ]) //
 )(i & 31), // initial call to h with i mod 32
 B = Buffer, // define B for Buffer
 g = i => // g is a helper function taking an integer
 s = 1065233 // representing an ASCII code minus 1
 >> i & 1 // and returning 0 for consonant / 1 for vowel
) //

Try it online!

s => // s = input string
s.replace( // replace in s ...
 /[a-z]/gi, c => // ... each letter c (case insensitive)
 ( //
 s + // if the previous letter was a vowel
 // (or this is the first iteration)
 g( // or the current letter
 i = // whose ASCII code - 1 is loaded in i
 B(c)[0] - 1 // 
 ) ? // is a vowel:
 "" // append nothing
 : // else:
 "ei" // append "ei"
 ) + ( //
 h = j => // h is a recursive function looking for the
 // replacement letter
 g(i) ^ // if the type of the current letter
 g( // does not match the type of
 j = -~j % 26 // the next letter in the alphabet  // obtained by incrementing j modulo 26
 ) ? // then:
 h(j) // keep advancing by doing a recursive call
 : // else:
 B([ // output the letter
 j + 1 | // whose ASCII code is j + 1
 i & 96 // with bits #5 and #6 taken from i
 ]) //
 )(i & 31), // initial call to h with j = i mod 32
 B = Buffer, // define B for Buffer
 g = i => // g is a helper function taking an integer
 s = 1065233 // representing an ASCII code minus 1
 >> i & 1 // and returning 0 for consonant / 1 for vowel
) // end of replace()

Commented

s => // s = input string
s.replace( // replace in s
 /[a-z]/gi, c => // each letter c (case insensitive)
 ( //
 s + // if the previous letter was a vowel
 // (or this is the first iteration)
 g( // or the current letter
 i = // whose ASCII code - 1 is
 B(c)[0] - 1 // loaded in i
 ) ? // is a vowel:
 "" // append nothing
 : // else:
 "ei" // append "ei"
 ) + ( //
 h = j => // h is a recursive function
 g(i) ^ // if the type of the current letter
 g( // is not equal to the type of
 j = -~j % 26 // the next letter in the alphabet (wrapping)
 ) ? // then:
 h(j) // do a recursive call
 : // else:
 B([ // output the letter
 j + 1 | // whose ASCII code is j + 1
 i & 96 // with bits #5 and #6 taken from i
 ]) //
 )(i & 31), // initial call to h with i mod 32
 B = Buffer, // define B for Buffer
 g = i => // g is a helper function taking an integer
 s = 1065233 // representing an ASCII code minus 1
 >> i & 1 // and returning 0 for consonant / 1 for vowel
) //