So this situation changes somewhat if you are talking about a natural weapon, like a claw, versus talking about unarmed strikes, because unarmed strikes (mostly) use the rules for manufactured weapons, not natural weapons.
Unarmed Strike
We’ll start with the simpler case of an unarmed strike.
If she already has the attacks, certainly
Each attack action is separate, unless something (like two-weapon fighting) says otherwise. Each time you choose to attack, you choose what weapon you are using. What weapon you use is entirely up to you.
Furthermore, adding or removing a hand from a weapon is a free action, so you can do that in between attacks in a full-attack. That means you can make your +6 attack with the bastard sword in two hands, let go of it with one, and attack with the free hand for your +1 attack.
This generally doesn’t serve very much purpose; usually one weapon is going to be better than another, so you’ll want to just make all your attacks with that one. But there can be exceptions; you could imagine, perhaps, an ability that triggers on an unarmed strike, but only once per round: attacking with an unarmed strike to trigger that, and then making the rest of your attacks with the biggest weapon available, may make sense.
The other issue here is that you can make unarmed strikes without using your hand; kicks, headbutts, whatever. That means you really don’t need to bother removing your hand from your biggest weapon. But maybe this hypothetical ability specifies you can only do it with a punch.
If she wants to gain extra attacks using two-weapon fighting, unclear
Someone with BAB +6/+1 just gets to make two attacks, no doubt about it. But someone who uses two-weapon fighting gets to make an extra attack, in exchange for a penalty to all attacks in that full-attack. But unlike the usual case where attacks can just be made with any weapon you like and they don’t influence one another, two-weapon fighting specifies that the attacks must be made with separate weapons, one “main hand” and the other “offhand.” The penalties you take depend on what that offhand weapon is. This is a harder situation.
In my mind, the way two-weapon fighting “works” is that both attacks are simultaneous, in-character (even though you roll dice sequentially, that’s just to keep things from getting confused). That means if you are attacking using both hands on the bastard sword, your hand isn’t free at the same time to make an unarmed strike. But again, you could maybe kick or headbutt or whatever simultaneously. And really, nothing says they are happening simultaneously in-character.
Paizo has issued an FAQ entry implying that non-punch unarmed strikes, and other weapons that seem to be hands-free like armor spikes, still somehow “take up” a hand, so that you cannot do this. These statements don’t make a whole lot of sense, aren’t well backed up in terms of what rules they are supposedly using for this claim, and in any event, don’t bear very much scrutiny. Nonetheless, the intent to prevent this is there, so Paizo, at least, considers it a problem to allow. Personally, I have not had any such problems.
Natural Weapons: even more unclear
Natural weapons are complicated and confusing, and the source of many, many questions on this site, particularly with respect to how they do or don’t combine with manufactured weapons. See this question for more detail about how they work.
Anyway, natural weapons don’t use “iteratives,” the extra attacks you get for having a high BAB. Someone who has BAB +11/+6/+1 attacking with just a claw only gets the one attack, at +11, and that’s it. If that same person attacks with a dagger in the other hand, the dagger goes first and uses up all the regular attacks, that is +11 and +6 and +1, and then the claw goes in addition to that, as a “secondary” weapon (which means a −5 penalty, i.e. +6, unless she has Multiattack).
Which means that it’s not clear that your character with +6/+1 could use the +1 for a claw even if she wanted to. She could certainly let go of the sword and have the hand free for the +1 attack, but mixing manufactured and natural weapons does not usually allow this; the natural weapon must come in addition, as a secondary attack.
The problem with that is that there is also a rule saying that you cannot use a natural weapon if that limb has already been “used” that round. By being the second hand on a two-handed weapon, it’s been used, so you can’t get to use it at all.
