With the help of my brother we've managed to do that. This is the solution:

Changes in XML:

<form string="Budget Line" js_class="copy_name">
...
<button name="button" button_class="button_copy_name" icon="fa-upload">

Whole JS file:

odoo.define('budget.button.copy_name', function (require) {
    "use strict";
    var FormController = require('web.FormController');
    var FormView = require('web.FormView');
    var viewRegistry = require('web.view_registry');
    var CopyFormController = FormController.extend({}, {
        _onButtonClicked: function (ev) {
            if(ev.data.attrs.button_class === 'button_copy_name') {
                // put concatenated strings into input
                // without change() you won't save inputed value
                // focus() is to put cursor into input field (i.e. to edit text immediately)             
                $("input[name='name']").val(`${ev.data.record.data.code} ${ev.data.record.data.name}`).change().focus(); 
            } else {
                return this._super.apply(this, arguments)
            }            
        }
    });
    var CopyFormView = FormView.extend({
        config: _.extend({}, FormView.prototype.config, {
            Controller: CopyFormController,
        }),
    });
    viewRegistry.add('copy_name', CopyFormView);
});