Please help to find grille clips

These may be "insert" window replacements/upgrades. An existing window has the jambs and parting bead removed. Then this unit is inserted and the original stop bead on the outside and inside holds this unit in place.

You can use the same idea.. trim the rough opening out so this window fits snugly. (Trim it like you would a door, with jambs and trim inside and, as needed, outside). Add stop bead on both sides of the window to cover the gap and hold it in place.

The Family Handyman has an article on How to Install Craftsman Trim, which says to simply glue and nail the stool to the jamb using 2 1/2" finish nails.

Once you have the stool cut to length, add a bit of wood glue to the length that will sit against the window frame. Position the stool where you want it, and drill pilot holes through the stool and into the jamb using a 5/64 drill bit (using a clamp to hold the stool in place can be useful). Next, fit 8d finish nails through the stool and into the jamb. Install the apron tight up against the bottom of the stool, making sure to nail into the framing when attaching it.

You could install a more substantial apron, and additionally nail through the stool and into the apron. This will add a bit more support for the stool (especially if the apron is attached to the framing), but may or may not fit aesthetically with the rest of the trim.

Obviously with this method you'll not be able to do chin-ups from the stool. But once the nail holes are filled and the piece is finished, it should look nice for years to come.