So with a bastard sword, you’re in luck, RAW—when kensai had you choose a single weapon, and you chose bastard sword, you chose something that is both a martial weapon and an exotic weapon. The definition of a martial weapon is a weapon you can use with martial weapon proficiency—and a bastard sword is that. The placement of the bastard sword in the exotic weapon table—rather than both, say—is not significant and there is no merit in reading more into that; that decision was almost certainly about saving page space.
If you are not convinced that bastard sword is a martial weapon, then consider this: a class grants Martial Weapon Proficiency as a bonus feat, or for some reason you choose to take that feat instead of Exotic Weapon Proficiency. Can you choose “bastard sword” for that feat? There are references to “Martial Weapon Proficiency (bastard sword)” in the rules, including in the very FAQ that’s been pointed to as a counterpoint, so that strongly suggests the answer is “Yes,” as does, ya know, the fact that the rules explicitly describe having martial proficiency in the bastard sword and what that means (and the fact that this proficiency existing is the very definition of a martial weapon). None of that could exist unless the bastard sword is a martial weapon. It’s just both martial and exotic. Anything that checks for martial status works with the bastard sword, and so also does anything that checks for exotic status, and they even work together at the same time—and that’s all fine, it causes zero problems under the rules.
It’s been this way since D&D 3.5e, and yeah, it could have been clearer then, and Paizo could have explained things better when they were adapting 3.5e to Pathfinder, but neither did and that shouldn’t stop us from understanding the thing correctly.
At any rate, kensai did not say that you gain “martial weapon proficiency” or “exotic weapon proficiency,” it said you gain proficiency with one weapon, that was either martial or exotic—or in this case, both. Your choice here was “bastard sword,” so you’re proficient in that weapon, which implies you’re proficient with all uses of it. That covers you for this purpose: you are proficient with the two-handed martial weapon, “bastard sword,” just as you are proficient with the one-handed exotic weapon, “bastard sword,” so that counts as one towards the student of war prestige class.
All that said, even if you had chosen some other exotic weapon, that wasn’t also a martial weapon, honestly it’d be a pretty terrible GM that said it didn’t count and you need to find another martial weapon. No one says someone with 20 Strength can’t take Power Attack because it requires “Str 13,” and this is the same situation—you have something that is better than martial weapon proficiency, so it should cover the martial weapon requirement. (We’ll ignore, for the moment, how rarely exotic weapons actually are better than martial weapons—the existence of the Exotic Weapon Proficiency feat, and the lack of classes that get proficiency in all exotic weapons, is proof enough of this claim, Paizo’s inanity aside. Also, for this exercise, it’s strictly true because the bastard sword does more damage than any martial one-handed weapon.)
This also explains the FAQ entry that obliquely touches on this situation—they say you get exotic proficiency because it is the better proficiency that lets you do more. I don’t think anyone is going to argue that the cleric described in that FAQ takes non-proficiency penalties when using the bastard sword in two hands. And this is, of course, the whole problem of using FAQs as stealth errata the way Paizo likes to do—strictly speaking, the FAQ is wrong. The cleric is said to get proficiency in the weapon, not some specific feat that gives proficiency in a particular usage of the weapon. But for the purposes of answering the specific question posed, it is an accurate, ish, answer. But does it now stealthily change how the cleric—and other things that use similar wording, like the kensai—work? It’s reasonable to wonder about that, and the fact that such a question even has to be considered is yet another reason why the FAQ is terrible, but the fact remains that extrapolating from the FAQ in that manner leads to nonsense results. So no one should rule that way—again, I’d rather say that any decent GM never would, which says something about any GM who does.
The point of that requirement is “simple weapons do not count.” It’s meant for “martial” characters, people who are serious about fighting with weapons. A kensai is that. Even having to get a second weapon proficiency is inane—the goal there, I imagine, was to keep out war clerics and the like, or force non-warrior characters to spend two feats rather than just one on Martial Weapon Proficiency. Kensai is not really like a cleric, and really ought not to have to jump through any more hoops to use that prestige class.
Finally, whether you need to jump through hoops or not, just fair warning—student of war is a poor prestige class, especially for a magus. It doesn’t progress your spellcasting, which means just taking more levels of magus will do you better than student of war will. Mind over metal and anticipate are pretty nice, but only “pretty nice.” The rest hinge off of know your enemy, which in most fights is going to amount to wasting a turn—and it doesn’t remotely give large enough benefits to justify doing that. If you were a fighter or something who wasn’t giving up very much, maybe you’d consider it for the occasional uses (though even then you should be able to find enough feats worth using to make it not worth it), but as a magus, you’re giving up a lot to get these only-sometimes-useful features. If you’re intentionally looking for something very low-power to match the party or campaign or whatever, sure, but if that isn’t your goal you should know what you might be getting yourself into here.