Have you tried a third solution which would be to create a new datepicker widget and add it to your grid to replace the "filled one" ?
here is a test where I do the "replace" in the submit handler but I guess it would be placed in a separate handler called 'submit again' for example...(but that is up to you) :
I changed your submit confirmation to show the date returned by the new datePicker (just to test)
//Opens the active spreadsheet and selects the first sheet.
var ss = SpreadsheetApp.getActiveSpreadsheet();
var sheets = ss.getSheets();
var sheet = sheets[0];
var popAttributes = {'padding':'10px','font-family':"Arial, sans-serif",'fontSize':'10pt','color':'#000099','background':'#ffffee','border-radius':'10px'}
var btnAttributes = {'padding':'3px','font-family':"Arial, sans-serif",'fontSize':'10pt','border-radius':'6px'}
function doGet() {
var app = UiApp.createApplication().setTitle('Test Form');
//Main panel with a 3x2 grid hosting labels and boxes.
var mainVP = app.createVerticalPanel();
var mainCP = app.createCaptionPanel('Purchase Details').setStyleAttributes(popAttributes);
var mainG = app.createGrid(3, 2).setId('grid');
var nameL = app.createLabel('Name');
var nameT = app.createTextBox().setName('name').setId('name');
var dateL = app.createLabel('Date');
var dateD = app.createDateBox().setName('date').setId('date');
var ownL = app.createLabel('Own/Lease');
var ownC = app.createListBox().setName('ownLease').setId('ownLease');
//Submit button with auto-reset functionality.
var subHP = app.createHorizontalPanel().setSize('100%', '40px');
var subL = app.createLabel('Submitted').setId('subL').setVisible(false);
var subB = app.createButton('Submit').setStyleAttributes(btnAttributes);
//Handler for button.
var subHandler = app.createServerHandler('submitHandler');
subHandler.addCallbackElement(mainVP);
subB.addClickHandler(subHandler);
//Add options to ListBox.
ownC.addItem('').addItem('Own').addItem('Lease');
//Place widgets on Grid.
mainG.setWidget(0, 0, nameL).setWidget(0, 1, nameT)
.setWidget(1, 0, dateL).setWidget(1, 1, dateD)
.setWidget(2, 0, ownL).setWidget(2, 1, ownC);
//Aligns the submit Button and Label to the right and adds them to their HPanel.
subHP
.add(subL).setCellHorizontalAlignment(subL, UiApp.HorizontalAlignment.RIGHT)
.add(subB).setCellHorizontalAlignment(subB, UiApp.HorizontalAlignment.RIGHT);
//Adds stuff to panels so it shows up.
mainCP.add(mainG);
mainVP.add(mainCP);
mainVP.add(subHP);
app.add(mainVP);
var doc = SpreadsheetApp.getActive();
doc.show(app);
}
function submitHandler(e) {
var app = UiApp.getActiveApplication();
//Set variables for each field in the Purchase Details CaptionPanel.
var name = e.parameter.name;
var date = e.parameter.date;
var ownLease = e.parameter.ownLease;
//I use a sheet.appendRow() method here but for the purposes of this question-
//we don't actually need the data, just need to clear it.
//Clears data from form to prepare for another submission.
app.getElementById('grid').setWidget(1, 1, app.createDateBox().setId('date').setName('date'))
app.getElementById('date') /* This is the field I need to clear somehow. */
app.getElementById('ownLease').setItemSelected(0, true);
//Displays the word "Submitted" when the Submit Button is clicked.
var label = app.getElementById('subL');
label.setVisible(true).setStyleAttribute('color', 'blue').setText(e.parameter.date);// to test the returned date value
return app;
}