How to convert a byte array to a string under a restrictive character set?

I would like to apply a cryptographic hash to an IP number and have the hash be in the character set "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789_".

My trials so far:

    String ipWithSecret = secret + "123.123.123.123";
    byte[] ipBytes = ipWithSecret.getBytes(StandardCharsets.UTF_8);
    MessageDigest md = MessageDigest.getInstance("MD5");
    byte[] mdBytes = md.digest(ipBytes);
    System.out.println("MD5:      " + mdBytes);
    System.out.println("US ASCII: " + new String(mdBytes, StandardCharsets.US_ASCII));
    System.out.println("Hex:      " + HexBin.encode(mdBytes));
    System.out.println("Base64:   " + Base64.encodeBase64String(mdBytes));        
    System.out.print("Binary:   ");
    for (byte b : mdBytes) {
        System.out.print(Integer.toBinaryString(b & 255 | 256).substring(1));
    }

outputs:

MD5:      [B@6d6f6e28
US ASCII: :???RN???z?}k^?
Hex:      3A8BB1881E524EDFEFE67AFF7D6B5E9D
Base64:   OouxiB5STt/v5nr/fWtenQ==
Binary:   00111010100010111011000110001000000111100101001001001110110111111110111111100110011110101111111101111101011010110101111010011101

Besides the hex, nothing is even remotely acceptable. My available alphabet is much larger than the hex one and, because I have to deal with a length limit of 16 characters, I would like to get hash string shorter.

Any suggestions how I can achieve this?

Answer I modified John's answer a little bit so credit goes to him. Here's my code that also handles collisions somewhat:

    String secret = "secret";  //config file, do not store in code
    Set existingMdBase64Alt = new HashSet(Arrays.asList("OouxiB5STt_v5nr_", "ouxiB5STt_v5nr_f"));
    String ipWithSecret = secret + "123.123.123.123";
    byte[] ipBytes = ipWithSecret.getBytes(StandardCharsets.UTF_8);
    MessageDigest md = MessageDigest.getInstance("MD5");
    byte[] mdBytes = md.digest(ipBytes);
    String mdBase64 = Base64.encodeBase64String(mdBytes);
    String mdBase64Alt = mdBase64.replace("+","_").replace("/","_" );
    System.out.println("Debug: " + mdBase64Alt.substring(0, 16));
    while (mdBase64Alt.length() > 16 && existingMdBase64Alt.contains(mdBase64Alt.substring(0, 16))){
        mdBase64Alt = mdBase64Alt.substring(1);
        System.out.println("Debug: " + mdBase64Alt.substring(0, 16));
    }
    System.out.println("Final: " + mdBase64Alt.substring(0, 16));