I only pay with dollars

Python 3.6 (pre-release), 87

lambda s:f"${int(s.translate({46:'',44:''})[1:])*[110,15,0,137][ord(s[0])%4]/1e4:,.2f}"

Uses f-strings to evaluate the result and format it.

s.translate({46:'',44:''}) removes dots and commas from s, thus making it a valid int literal, then int(...) converts it into the actual int object.

Convex, 56 55 54 bytes

(\®\.|,"ö)\e_\'.\++~\"€£¥"#[1.1_.27+.15]=*"%,.2f"\Ø'$\

Well, this can definitely be shortened. Try it online!

Saved a byte thanks to Lynn!

Explanation to come when I can get access to a computer.

Java 7, 240 227 215 211 207 202 199 196 bytes

(201 - 2 bytes because of the rule "If you use one of the above symbols (€, £, ¥), you may count them as 1 byte")
Thanks to @Frozn for saving a lot of bytes.

String c(String a){int c=a.charAt(0);return java.text.NumberFormat.getCurrencyInstance(java.util.Locale.US).format(new Long(a.substring(1).replaceAll(",|\\.",""))*(c<'¥'?1.37:c>'¥'?1.1:.15)/100);}

Ungolfed & test code:

Try it here.

class Main{
  static String c(String a){
    int c = a.charAt(0);
    return java.text.NumberFormat.getCurrencyInstance(java.util.Locale.US)
        .format(new Long(a.substring(1).replaceAll(",|\\.","")) *
                 (c < '¥'
                   ? 1.37
                   : c > '¥'
                     ? 1.1
                     : .15
                  ) / 100);
  }

  public static void main(String[] a){
    System.out.println(c("€1,37"));
    System.out.println(c("£4.00"));
    System.out.println(c("¥2,782,122.78"));
  }
}

Output:

$1.51
$5.48
$417,318.42