oak table restoration question -- how to better apply edging over endgrain?

I'm trying to restore an oak table. The top is about an inch thick and is split right down the glue joint, which will be easy enough to fix.
The more complicated issue is the edging. The original builder put rounded edging (an inch thick or so) right around all four sides. It appears to be supported by wooden strips recessed into the bottom of the table and the edging, nailed in place with brads. With the grain it appears to also be glued, but along the endgrain it is loose.
Any suggestions on how to fasten this so it doesn't sag whenever anyone leans on the end edging?
I was thinking about insetting some dowels from underneath near the sides, then gluing the middle and screwing through the edging into the dowels with elongated holes to allow for seasonal movement. I'd have to either fill with putty or else plug and try to match the original finish. I really don't want to refinish the whole table at this point if I can avoid it.
Ideas are welcome.

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