The easiest way would be to add
<input type="submit" name="update_cart" value="Update Cart" /> to your template and then add a bit of jQuery to bind the update link to the newly created update button (which you would want to hide to keep your current design).
As long as your submit button isn't called
commit (both of which submit the order),
empty_cart (which empties the cart) or
next (which updates the cart and redirects the customer to the url specified in your
next_url parameter), Store will simply update the cart quantities and redirect you to the current page.
This allows you to have both "Continue to Checkout" and "Update Totals" buttons on the same form, which both update the cart but redirect the customer to different locations. This can be seen in the
checkout.html example template which ships with Store:
<div class="form-actions"> <input type="submit" name="update_cart" value="Update Totals" class="btn" /> <input type="submit" name="empty_cart" value="Empty Cart" class="btn btn-danger" /> <input type="submit" name="next" value="Continue to Billing Details" class="btn btn-primary" /> </div>