Hi, I'm an astrophysicist, I am well aware of the IAU (International Astronomical Union) definition of a planet. It is a definition that is relatively controversial and doesn't really make a lot of sense - it was mostly written to prevent the list of planets from getting too long as we discovered more dwarfs. This definition does not include anything to do with barycentres - I'll get back to that.
The IAU definition of a planet has 3 parts:
1. Object in orbit around our Sun
2. Object has reached hydrostatic equilibrium (i.e. it's a sphere, not a potato)
3. Object has cleared its neighborhood around its orbit
Both Earth and Luna satisfy 2. I will accept that 1 is debatable, but the crux of my argument is that Luna is the size of a terrestrial planet (yes, ~1% of Earth's mass, but still huge and of roughly the same order of magnitude as Mercury, which is a mere ~5% of Earth's mass) and exceptionally large relative to its orbital partner compared to any other "satellite", a factor the IAU does not account for.
It's important to note that 1 literally does not permit binary planets - even if both bodies are identical, one must be the "moon". This is major criticism of the IAU definition. It also doesn't account for exoplanets or rogue planets, but that's another story.
3 is where this definition falls apart. You could easily argue that most of the planets in our solar system fail depending on how you interpret it, because it's so fuzzy. The Earth fails, because there's a planet-sized body chilling out in its orbit (Luna), Jupiter fails because it has huge quantities of asteroids trapped in its Lagrange points, etc.
In the latter case, we say that because Jupiter is determining the motion of these bodies, it still counts. For the Earth we simply ignore Luna and say that the rule is more about stray bodies than orbital partners. But by the same logic we can say that for Luna we ignore the Earth, and Luna passes.
So I would argue that the IAU definition is bad, but if you fixed it so that it allowed binary systems to exist, it would readily define Luna as a planet in my view (if you deleted the Earth from the solar system and left Luna, it would unambiguously meet the criteria).
Now let's talk about barycentres.
The barycentre of two bodies in orbit is just their overall centre of mass. People often point to the external barycentre of the Pluto-Charon system to indicate that they are binary dwarfs. However, this is a poor metric in my view, because it's highly dependent on orbital distance.
The centre of mass of the Earth-Luna system is inside the Earth at present, but if Luna simply orbited further away, the barycentre would be in the empty space between them, with nothing about the two bodies individually, or the qualitative nature of their orbits, having changed. In fact, given enough time and pretending the Sun won't consume both bodies in a few billion years, Luna would actually drift far enough away due to tidal interactions for this to happen - it would be silly for it to suddenly be a planet one day when it wasn't the day before.
To answer your question, no. I would under no circumstances argue that the martian moons are planets. They are minuscule space potatoes that are not even large by the standards of asteroids. If Mars were close enough in mass to them for the system to be considered plausibly trinary, it would be far too small to even qualify as a dwarf planet (and there's no way such a system would be gravitationally stable to perturbation by Jupiter regardless).
