Accept the invitation, explain that your friend has left, and provide an "out" in case she's not interested in hanging out with you, alone.
Although it's good to be considerate of others, this is a pretty straightforward situation. And fortunately the information you want to convey isn't a big deal. The following assumes that you're willing to spend time with her, as suggested in a comment under the question.
First, if you don't want it to appear that you are rejecting her invitation, don't reject it. It's that easy.
Second, answer on behalf of your friend who cannot attend due to having already left the area. Since the invitation was for both of you, it makes sense for there to be an answer for both of you. Because your friend has already left, you're not making a decision for him or anything like that, you're just explaining that he will not be able to attend.
Third, if you're truly concerned that this will change her decision, offer her an "out" to cancel the plans. Doing this step last allows you to clearly accept the invitation while still giving her the information she's lacking. If she's as unsociable as you seem to think, mentioning something about plans changing or being cancelled can help make it clear that that's an acceptable occurrence-- she may feel less pressured to go through with the hangout if she doesn't want to do it one-on-one. But this is her decision to make, not one for you to make on her behalf.
Thanks for inviting me, [acquaintance], I'd love to! [Friend] won't be able to come along, though, since he's already left. I'll plan on meeting you at [place] around [time], and let me know if anything changes.
Why is this approach a reasonable one?
My main thought is that, if you don't know her very well, it's not reasonable or appropriate for you to assume strong knowledge of her mental state or desires and then make her decisions for her. Refusing to spend time with her because you assume she'll feel awkward in your company, even when she's requested your company, is odd.
Her not being a very social person (in your estimation) is a red herring. She has initiated this proposed event, and so it's fair to presume that she wants your company. For all you know, her unsociability comes from shyness, and it was a big deal for her to invite you to socialize, and she would be every bit as happy (if not necessarily less awkward) to spend time with just you as with you and your friend both.
You two not knowing each other very well also doesn't seem as significant to me as you seem to feel. People don't generally meet and immediately become best friends, but rather their relationships build (or don't) over time. Saying that you aren't friends, and so you don't want to spend time with her, suggests that you don't want to become friends with her. There's no way to soften that impression.
I don't quite understand your concern about it seeming too much like a date. If you're both on the same page and specifically don't want it to be a date, it's not going to be a date.
I've made plans with groups of people and individuals, and had those plans change at various times due to unavailability/surprise availability in ways that have suggested date -> not-date, not-date -> date, spending time with people I like, spending time with people I don't like, and every unclear situation in between. It's just not a big deal, with a single exception:
When plans narrow from [person A] and [me + others] to just [person A] and [me], and they cancel those plans, it makes me feel bad. Even in cases where one of the others was the explicit point of getting together (such as a case where I'm a "wingman" for a friend trying to start something romantic with someone), it's made me feel unliked and unwanted. They'd already planned to go out, but if it's only to see me they'd rather do anything else (including nothing). That sucks.