I also used a fishing rod holder mount
for my Magma Newport on a previous boat. The upside to a rod holder mount is installing one doesn't look out of the ordinary on a boat. Know that there are rod holders designed for rail mounting. That's what I used on a horizontal stern rail.
One of these, I think:
F16-2600: Stainless Steel Clamp-on Adjustable Rod Holder | TACO Marine
I eventually moved to using Magma's detachable post
with a ring mounted to the deck. I've seen others use a Garelick recessed mount, the kind used for 3" table legs.
One question is do you want a permanent or removable mount? Either way perhaps you can find a local fabricator to create a metal bracket that clamps more securely to the rails. Maybe bridge between two of them if you're looking for something permanent.
There was another thread here on this: