I do not think T&G boards alone are best for this kitchen because of fire risk. It might even violate the code. At the very least you should put drywall first and then the t&g boards over that.
But a much better solution would be to forget the t&g boards and instead put on a "tin" ceiling over drywall. This would add the design you want with a very high level of fire protection.
Modern metal panels for ceilings come in a wide variety of finishes and have several different attachment methods. My wife wanted a tin ceiling in an 8' x 8' vault in our kitchen and it was not that hard to DIY. The type she bought was 2'x 2' panels from American Tin Ceiling, but with a design that makes it appear to be 1' x 1' panels.
If your ceiling joists are 2' on center, these 2x2 panels would be perfect fit. You would put the cross strapping on as you planned and then attach more of the same strapping to the exposed bottoms of the ceiling joists to make a square grid of strapping. Put on drywall, tape and bed to firewall grade, then nail the metal panels through the drywall into the strapping. The metal panels are only nailed around the perimeter.
It is possible that the right mastic would allow attaching the panels to the drywall without nailing. The panels are very light.
I followed the mfgr recommendation for our tin ceiling and used stainless steel 18 ga brad nails with a pneumatic nailer. Adjusting the air pressure allowed precise driving of the brads so they are exactly the right depth.
EDIT
An alternative to strapping is to attach plywood (~ 1/2") on the underside of the ceiling joists, then drywall, then the metal panels. This would allow exact centering of the metal panels in the space. I attached two 4x8' sheets of near 1/2" plywood on top of drywall which was already there. Then I nailed the metal panels to the plywood with 3/4" or 7/8" long 18 ga stainless brads.
Exact centering of the pattern in the metal panels becomes important in positioning ceiling light fixtures. From the living space side you want the fixtures to be symmetrically located in the kitchen ceiling, but I also wanted each of our four fixtures in the center of one of the four squares that make up each panel. I used a keyhole saw on a drill to cut the holes for the four boxes. In retrospect one modern multiple LED fixture in the center of the space might have been better than the four we put in.