Your server will be sending a JSON:

my $PDF = ...; # read PDF binary contents from file
my $LOGSTR = ...; # read LOG file contents
use MIME::Base64;
my $data = {
  # this is binary that's why we encode it to fit in a JSON string
 'pdf' => encode_base64($PDF),
  # this is text but it does not harm:
 'logstr' => encode_base64($LOGSTR)
};
# send the data to client:
# CGI version:
use CGI qw(:standard);
use JSON;
print header('application/json');
my $json_text = to_json($data);
print $json_text;

# OR render JSON to go to client with Mojolicious
my $c = ... # controller
...
$c->render(json => $data); # $data as above
[download]

Here is javascript for the client (more like untested hints):

function ajaxit(
  url, // server endpoint
  method, // e.g. POST
  data // any data to send to the server for this request?
){
  var xmlhttp = new XMLHttpRequest();
  xhr.open(method, url, true);

  // this is the function which handles the xhr data transfer
  xhr.onreadystatechange = function () {
    // xhr.readyState can be 0 to 4 and each one comes here
    // 4 means transaction is all done,
    // 3 means data is received.
    // for error or success, state is 4
    if( xhr.readyState !== 4 ){
      // data is being transfered still, wait ...
      return;
    }
    // xhr.readyState=4, all is done, either on error or success:
    if( xhr.status === 200 ){
      // data was received, make it a JS object (basically a hashtable
+ aka JSON)
      var data = JSON.parse(xhr.responseText);
      // check integrity here
      // ...
      // handle the data your html must have a div to put the <a>'s
      // with this adivID
      json_data_handler(data, adivID);
    } else {
      // we did not get a HTTP code 200 OK,
      // if we have a callback to handle errors, call it
      if( onError == null ){
        console.log("ajaxre() : error, "+method+"ing to '"+url+"':\n  
+readyState="+xhr.readyState+", status="+xhr.status);
      } else {
        # handle failed request
        #onError(xhr.readyState, xhr.status);
      }
    }
  };

  // now do the XHR request with caller-supplied data
  xhr.send(data);
  // this call will be handled by the function above
  // function is asynchronous and leaves here while request is being m
+ade
}

// this is called when XHR was successful and we got data from server
// that data was converted to a JS object (see above ajaxit())
// and this data is given here as dataobj
// your html should contain a DIV somewhere (whose id you supply)
// to add two <a> tags in there for the user to click
function json_data_handler(
  dataobj,
  divID
){
  // we have already sent an XHR request to server
  // and it gave us back data (as a xhr.responseText)
  var obj = JSON.parse(data);
  // do here some checks if this was successful
  // ...
  // atob decodes base64 strings to bytes,
  // var pdf is now the actual binary pdf content
  var pdf = atob(dataobj['pdf']);
  var logstr = atob(dataobj['logstr']);

  // client-side save-as filename, whatever
  var saveAsFilename = '...';

  // do whatever with pdf data
  var el = document.getElementById(divID);
  var a = document.createElement("a");
  el.appendChild(a);
  # edit: set the right headers for pdf data
  #var blob = new Blob([pdf], {type: "octet/stream"});
  var blob = new Blob([pdf], {type: "application/pdf"});
  # edit: setting an intermediate variable which should be
  # revoked when not needed anymore with
  #    window.URL.revokeObjectURL(objurl);
  var objurl = window.URL.createObjectURL(blob);
  a.href = objurl;
  a.download = saveAsFilename+'.pdf';

  // do the same for logstr
  // or, since it is text, display it in the div verbatim
}
[download]

above is totally untested but the idea was tested

Apropos processing latex from Perl, I always use LaTeX::Driver to run latex.

bw, bliako