Why did he not let Harry - whom IIRC we previously saw saying that Hermione ought to be sent to Beauxbatons - beg Hermione to leave, or failing that, order her?

"Why did he not let"? I don't see any place where Quirrel isn't "letting" Harry do these things. Perhaps your question should be better phrased why he didn't ask Harry to do these things?

I think a simple enough answer is that he feels he has a better chance of convincing Hermione to leave, than to convince Harry to force Hermione to leave against her will. Since Bellatrix, Harry has learned to inquire about what is in it for Quirrel when Quirrel asks him to do things. And he'll see that wanting Hermione to leave may be to the advantage of whomever wanted to frame her in the first place, as both events lead to a Hogwarts without Hermione in it.

Is this all reverse-psychology?

I don't understand your usage of the term. It's me who's saying he wanted her to leave (aka non-reverse psychology), it's LKtheGreat and you who seem to be saying he was applying reverse-psychology and that he really wanted her to stay.

Why did he make the blatantly manipulative hard-sell tactic of 'buy now, this is a limited-time offer only!'

The simplest explanation of why someone tries to manipulate you into doing something is because they want you to do it.

And frankly he came very close to getting her to say "Yes." We were inside Hermione's head. Quirrel came close to succeeding. If someone comes that close to succeeding, and fails just by something tiny which is outside their control, then the simplest explanation is that they wanted to succeed.