Is there a GHC extension for enabling overloaded character literals?

Not as of GHC 8.8, but you can use the QuasiQuotes extension to get pretty far. Here is an example of a quasiquote which only accepts an ascii character and converts it to its byte representation.

import Language.Haskell.TH
import Language.Haskell.TH.Quote
import Language.Haskell.TH.Syntax

import Data.Word (Word8)
import Data.Char (isAscii)

asciiByte :: QuasiQuoter
asciiByte = QuasiQuoter
  { quoteExp = \str -> case str of
                         [c] | isAscii c -> lift (fromIntegral (fromEnum c) :: Word8)
                         _ -> fail ("asciiByte: expects a single ascii character, got " ++ str) 
  , quotePat = \_ -> fail "asciiByte: only available for expressions"
  , quoteType = \_ -> fail "asciiByte: only available for expressions"
  , quoteDec = \_ -> fail "asciiByte: only available for expressions"
  }

Then, you can use it as:

ghci> [asciiByte|a|]
97
ghci> [asciiByte|é|]

<interactive>:75:12: error:
    • asciiByte: expects a single ascii character, got é
    • In the quasi-quotation: [asciiByte|é|]
ghci> [asciiByte|abc|]

<interactive>:76:12: error:
    • asciiByte: expects a single ascii character, got abc
    • In the quasi-quotation: [asciiByte|abc|]

No.

Let me answer that with some exploration of the GHC source code:

This is the code in the renamer where OverloadedString take effect:

rnExpr (HsLit x lit@(HsString src s))
  = do { opt_OverloadedStrings <- xoptM LangExt.OverloadedStrings
       ; if opt_OverloadedStrings then
            rnExpr (HsOverLit x (mkHsIsString src s))
         else do {
            ; rnLit lit
            ; return (HsLit x (convertLit lit), emptyFVs) } }

rnExpr (HsLit x lit)
  = do { rnLit lit
       ; return (HsLit x(convertLit lit), emptyFVs) }

rnExpr (HsOverLit x lit)
  = do { ((lit', mb_neg), fvs) <- rnOverLit lit -- See Note [Negative zero]
       ; case mb_neg of
              Nothing -> return (HsOverLit x lit', fvs)
              Just neg -> return (HsApp x (noLoc neg) (noLoc (HsOverLit x lit'))
                                 , fvs ) }

(https://github.com/ghc/ghc/blob/c2991f16cb6f5b4e7cff46a394dda4247d973f44/compiler/rename/RnExpr.hs#L152)

You see that there is special handling for strings, but not for other forms of literals. The last clause is for literals that are overloaded according to the original Haskell standard, as we can see in these lines from the parser:

        | literal                       { ECP $ mkHsLitPV $! $1 }
-- This will enable overloaded strings permanently.  Normally the renamer turns HsString
-- into HsOverLit when -foverloaded-strings is on.
--      | STRING    { sL (getLoc $1) (HsOverLit $! mkHsIsString (getSTRINGs $1)
--                                       (getSTRING $1) noExt) }
        | INTEGER   { ECP $ mkHsOverLitPV (sL1 $1 $ mkHsIntegral   (getINTEGER  $1)) }
        | RATIONAL  { ECP $ mkHsOverLitPV (sL1 $1 $ mkHsFractional (getRATIONAL $1)) }

(https://github.com/ghc/ghc/blob/c2991f16cb6f5b4e7cff46a394dda4247d973f44/compiler/parser/Parser.y#L2793)

…

literal :: { Located (HsLit GhcPs) }
        : CHAR              { sL1 $1 $ HsChar       (getCHARs $1) $ getCHAR $1 }
        | STRING            { sL1 $1 $ HsString     (getSTRINGs $1)
                                                    $ getSTRING $1 }
        | PRIMINTEGER       { sL1 $1 $ HsIntPrim    (getPRIMINTEGERs $1)
                                                    $ getPRIMINTEGER $1 }
        | PRIMWORD          { sL1 $1 $ HsWordPrim   (getPRIMWORDs $1)
                                                    $ getPRIMWORD $1 }
        | PRIMCHAR          { sL1 $1 $ HsCharPrim   (getPRIMCHARs $1)
                                                    $ getPRIMCHAR $1 }
        | PRIMSTRING        { sL1 $1 $ HsStringPrim (getPRIMSTRINGs $1)
                                                    $ getPRIMSTRING $1 }
        | PRIMFLOAT         { sL1 $1 $ HsFloatPrim  noExt $ getPRIMFLOAT $1 }
        | PRIMDOUBLE        { sL1 $1 $ HsDoublePrim noExt $ getPRIMDOUBLE $1 }

(https://github.com/ghc/ghc/blob/c2991f16cb6f5b4e7cff46a394dda4247d973f44/compiler/parser/Parser.y#L3708)