So it then comes down mainly to Ajax's agility, which is a bit of a moot point as he doesn't appear to be much quicker than Lust or Gluttony, both of which were hit fairly easily.
Were either of these faster than Guts? Because I remember Guts being described as incredibly fast, and it was decided that Ajax could keep up with him.
we know that Ajax was not superhumanly invulnerable.
Actually, this is untrue. If you check the 'Death of Ajax' section
here, it will tell you that Ajax had to jump on his sword so that it entered his armpit, his only vulnerable part. Also,
here, you will find that a well-made bronze Xothos actually bent, instead of entered Ajax's skin. Bronze might not be as strong as steel, but it's still pretty impressive to bend a bronze rod against your abdomen.
Let's start off with Ajax's shield. It's made mostly of cow-hide. Leather that could be easily burned away by Mustang. Bronze is much tougher, but I think it could be heated to the point that if it didn't melt, Ajax would have to shed it just to keep his arm intact.
Cow hide is actually harder to burn than you'd think. According to Lord Shonus, it ignites at 212*F when dry. When wet, as was the practice of Greek soldiers (They had a whole ocean to do this in) it becomes very difficult to light a seven-cowhide-thick wet piece of skin on fire. (In fact, a wet cow hide has an ignition point of 2120*F) Wetting a leather shield increases the tensile strength of a shield as well as increasing its resistance to the pitch-covered arrows of their enemies. It does make it heavier, but that's not much of an obstacle for Ajax. According to
the explanation of the Apollo disaster, a pure oxygen fire reaches around 1850*F, a temperature that may evaporate some water, but not ignite the leather. Because Roy's alchemy is instantaneous, Ajax could keep diving underwater to soak it more. Roy can't harm Ajax underwater because of the large amount of moisture, according to
this.
Asides from this, Ajax could merely play it defensively until he gets within range of Roy, by scrunching his arms up to hide his pits. Ajax would likely last longer endurance-wise, so he could just stand on the beach, somewhat submerged in water to keep wet. Ajax has held his breath (pushing a ship ashore) for hours at a time, so he could wait until Roy was exhausted. Ajax also is more agile than the average man, with the ability to leap between ships with a single bound and punt Trojans with a 20 foot pole while doing so.