Here's how I did it. Expect some frustration. I'll list all the steps, including the mess-ups, so you can see what you're looking at and what not to do. I realize I've very late to this party, but I know this is an ongoing issue.
Buy thin rif-cut veneer with paper backing. Mixed oil-based stains to closely match original color. Don't like (actually wife doesn't like) rif-cut pattern.
Order flat-cut veneer. Much nicer pattern. But paper backing is much thinner, once it's stained, glue won't stick to it (it's like trying to glue paper soaked in oil). Start over with water-based stains, find a mix of stains that closely matches original color (see photo). Glue it on, find that chosen glue makes veneer buckle even with clamping. Rip it all off, sand off glue.
Cut/stain another piece of veneer, use contact cement to attach. This one works.
Hints: Cut veneer a little large and trim it down. Trim AFTER staining. Veneer is normally glued in a vacuum press, so some glues will require clamping with one or more large pieces of wood to keep it flat. Test glue on scrap material to be sure it will work. You'll have to sand the damaged surface for any glue (including contact cement) to stick well. Best to sand it down to bare particle board. I finally settled on contact cement because I wasn't confident I could get good clamping force across the entire surface. I didn't have much luck when I tried construction adhesive.
Plan on spending several hours on this over multiple days.
You might find that the iron-on veneer is easiest to work with. I couldn't find any the right size in white oak, but other woods (including red oak) are available with the iron-on glue. Also, there are iron-on glues that you can apply to the surface but I didn't try them.
My total cost was north of $150, but part of that is the purchase of the veneer twice.