Anyway, I think the bit about them not being able to cast spells on each other (which is true-ish in canon) is a stronger argument.

Probably, but EY is a tricky writer and can make me second guess everything.

It appears that the phrase "The sudden sense of doom clashed with Harry's steel determination", tells us what happened. The Quirrell doom field brought Harry to his senses. In that context "WRONG DON'T" appears to be Harry's awakened response to what he is doing. Nothing seems to imply that Harry's will is not his own.

However, the sense of doom suggests a connection between Harry and Quirrell. Based on canon this suggests Q=V, but I suppose it might also suggest that Quirrell has been Voldemort. The connection with Harry could be residual.

That special connection could be the source of "WRONG DON'T". It implies that Quirrell has a subtle route to Harry's mind that does not require the Imperius curse and that gets around Harry's Occlumency. In canon Harry could sense Voldemort's mood and occasionally see through his eyes.

So perhaps Quirrell is reading Harry's mind and carefully manipulating him through the scar connection. This would help to explain Quirrell's ability to make deductions from insufficient evidence.

Chapter 49, Prior Information:

There were times when Harry suspected that Professor Quirrell had way more background information than he was telling, his priors were simply too good.