I'm being asked to reproduce this panel material to a different spec than this sample. The sample is 1" x 3/4" solid white oak spaced 5/8 and glued in 1/8" grooves in a 3/4 MDF panel with white oak veneers.
The customer wants 1" x 7/8" solid, spaced 1/4", same 1/8" groove. They are also concerned about the MDF substrate because it will be installed in a bathroom, and want a different material. Makes sense to me. Will have a clear poly finish.

My question: what type of panel core would be a good choice if not MDF? I'm concerned that with so much material removed from the top of the panel, the replaced with solid, it will become unstable and start to bow/warp/taco?
In the bathroom, most of the panels are 2' - 3' wide. There is also a master bedroom that would use full 4x8 panels, but less of a moisture issue.

I've read here in other Woodweb forums about an "MDX" type panel that uses more waterproof binders, but haven't seen it for sale from any of my my local supplies.

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My first suggestion would be to reduce the back panel thickness to 1/2". This will help reduce the warping forces. I don't think the warping will be a great issue if the panels are solidly screwed to the framing. Also I would not glue the sticks into place, screw them on centre. This will allow the sticks to expand and contract without imparting that dimensional movement to the panel as a whole.
Be aware of what the wood moisture content is and what the predicted highest MC will be when you dimension your stock and groove. The sticks will no expand much individually but you don't want to strain the shoulders of the grooves.
There is nothing wrong with industrial particle core for the substrate for this application. Just spray the panel fully front and back before attaching the sticks.

As stated above, screw the solid pieces in the groove with no glue. I think i would even put a shoulder on the solid pieces so the varying amount of shrinking and growing is not a concern, nor visible.

I can see the advantage in screwing, but that's a lot of assembly for 350 sq ft when I could just put them in my press. Just noticed my photo above appears squished and distorted... here's another I think it will show the actual dimensions. Will there really be that much expansion in a 1" wide piece?

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The rationale for not gluing the slats in place is to allow the slats to expand and contract independent of the back panel. I agree it would be much faster to just glue and press them. I would then just run a minimal line of glue down the centre of the groove.

I was also basing my statement off the skewed photo and not the 1" 7/8" description. Glueing them up in the groove should be no problem.

Well I contacted the company that makes the original product, turns out they screw their oak to the MDF! I can't imagine how they do that at a production scale, but perhaps that's why I've been asked to recreate it cheaper and faster than the company can supply it.
I'm still in favor of gluing, partially because I'll be making a number of odd sized sheets to fit around vanities and such, I'd rather not worry about screws getting in the way of those cuts.
I think what I'll do is cut some two small kerfs into the bottom of the oak stock about 3/8 apart, which should help the glue to stay in the center of the groove.

I was constantly bombarded with, "Surely you can make these cheaper" requests. NO, I can't make it cheaper than a production factory! Be very cautious with your bid. With the course grain of oak, I'd present a sample of using 23 gauge headless pinner. Especially if they are making this an economy version.

I consulted on a job similar to this for elevator panels only the back panel was painted aluminum. We designed a comb jig to hold the sticks and the panel was lowered onto the jig and sticks. The back panel was pre-drilled for screws.
Consider getting a quote for the sticks to be supplied D4S to you. That is if you don't have a moulder or 4 sided planer. It is a huge labour savings and surprisingly inexpensive.
I see by the sample that the sticks are oriented with the rift/quartered face up. This will halve the seasonal movement if you are concerned about expansion and also look better in my opinion.

If your confident in your finish I'd just plow out tight fitting dado's (CNC hopefully) in veneer core material and install the battens however you choose (scews/staples/etc) and plow the finish on. Even if it breaks in the corners it will be so minimal/long term it will never likely be an issue.

If they are hell bent on bullet proof plow the dados oversized for finish, prefinish all the battens and prefinish the panels and screw/staple the battens on from the back.

I recently bought some MDF to make window sills with. Supplier said it was "waterproof" I put a piece of scrap in a jar of water. It's been there for over a month and has done nothing. No swelling, no dimension change, nothing. It's made from rice straw. I though interesting! Going to do some exterior columns from the same stuff. I'd think it would work for your bathroom project.