Antique furniture repair question

I keep seeing my photos upside down and can't find a way to correct this. Please let me know how to change this.
I'm trying to repair an antique mahogany dresser missing a trim piece separating two drawers. The original trim piece was lost during various moves and was already broken at the end.
As you can see in the first pic of the drawer there is a gap between the bottom and middle drawer. This trim piece is 3/4" wide and originally connected with a mortise into the leg . It looks like the tenon broke off inside the mortise (second and third pic). The forth pic shows what the joint looks like on a matching dresser. The mortise-tenon joint seems to be a small vertical oriented joint and don't seem very strong. I think an additional problem is there is no way to support the trim piece over it's 39" length.
I've scratched my head over this too many times (I don't have much hair as it is:rolleyes:). I have a piece of mahogany (seen in the first pic) with similar grain and I found a stain that matches the existing color of the dresser.
I don't know how to insert a strong joint that will last. My latest thinking is to drill out the old mortise and glue a larger tenon into the leg. I would then make a rabbet in the back underside at each end of the trim piece to glue into the tenon. I'd also add braces in each corner behind the trim piece.
This still doesn't seem like a durable repair that will last another 100 years.
Sawmill Creek has bailed me out of several disasters and I've always appreciated your ideas. Any suggestions?
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Drill -dowel -rabbet -brace - just as you describe is exactly what I was thinking. FWIW.

First off while your thumbnails are inverted they right themselves when opened. (At least for me). Can you put a horizontal piece of say 1/4" plywood between the drawers (I think it is referred to as a dust cover?) to help support the trim piece? Cut a matching rebate in the back of trim piece and glue it in.

What photos Gary

Quote:

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Originally Posted by Hicks Benedict

What photos Gary

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Hicks - I believe that you need to become a SMC contributor to see the photos. Cost is $6 per year.
David

Whenever I have an antique repair issue. I go to YouTube antique repair guru, Thomas Johnson. Amazing content

I recently repaired an old dresser that was literally falling apart. I cleaned the old glue from mortise and tenons and made a couple of parts that were missing. The joints were very loose so I used thickened epoxy. It is very usable now. Adding the dust divider is a good idea. It may have originally had one.

Stain tip is to try iodine to mach that old yellowed look.
Bill D

Thanks for all the helpful comments. I've been a long time contributor to SMC so it's more an issue of my iPhone and not having the right app. I've used this phone to post pics before and never had a problem.
I thought about using a dust cover as additional support but there's little room for this. I'm going to give this another look, hopefully tomorrow when I can get back into the wood shop.
I'll look into iodine. I have painful memories from having iodine applied to childhood wounds.:)