Same number letters

R, 106 bytes

function(s){for(A in L<-LETTERS)s=sub(A,strrep(A,max(x<-+s-+Map(gsub,L,'',s,T))-x[A]--1),s,T);s}
"+"=nchar

Try it online!

Base R approach :

• stealing some ideas from @J.Doe R+stringr approach, I saved 26 bytes !
• another 5 bytes saved using @J.Doe suggestion to abuse R + operator

Perl 6, 82 bytes

-3 bytes thanks to nwellnhof

->\a{a.=lc.=subst($_,$_ x a.comb(/<:L>/).Bag.values.max+1-a.comb($_))for 'a'..'z'}

Try it online!

Takes a mutable string and modifies it in place.

Explanation:

->\a{        # Anonymous code block that takes a mutable string            }
 a.=lc;  # Lowercase
                                                               for 'a'..'z'  # For each letter
 .=subst(                                                    )  # Substitute
          $_,   #The first occurrence of the letter with
             $_ x  #The letter repeated
                  a.comb(/<:L>/).Bag.values.max    # The count of the most common letter
                                                 +1  # Plus 1
                                                   -a.comb($_)  # Minus the count of that letter already in the string

05AB1E, 16 bytes

lDáÙSйls¢Zα>×.;

Try it online!

Explanation

l                  # convert input to lowercase
 D                 # duplicate
  á                # keep only letters
   Ù               # remove duplicates
    S              # split to list of chars
     Ð             # triplicate
      ¹ls¢         # count the occurrences of each letter in lowercase input
          Zα       # absolute valuue with max occurrence
            >      # increment
             ×     # repeat each unique char that many times
              .;   # replace the first occurrence of the char in lowercase input with this