Some more thoughts on this possibility from a purely analytical perspective: The safer play for Trump would seem to be limiting Mueller’s ambit to keep his investigation from running out of control as Andy has suggested, political warfare against him in preparation for discrediting any damaging findings (see Bill Clinton v. Ken Starr), and using both “the everyone does it” defense (again, see Bill Clinton) and the pardon power when and if necessary.
Maybe the rumblings from Trump on Mueller are a way to pressure Rosenstein (i.e., rein in Mueller or this thing could get really ugly), which might have some effect, but I don’t think Mueller is going to be swayed by any “brushback pitches.” We can be certain that Mueller hates Trump for the same reason that Louis Freeh and Ken Starr hated Bill Clinton — straight-arrow law-enforcement types have a natural distaste for politicians who think they can talk or maneuver their way out of anything.
The only reason firing Mueller makes sense on Trump’s terms is if Trump finds it so intolerable that this investigation has reportedly spread from a Russia-collusion story to his business dealings that he simply can’t let it go, either because he thinks it’s so unfair or because he has things to worry about in his past. Even if it’s the latter, it would still be better to see what Mueller comes up with, and resort to the “old news” or “everyone — or every real-estate mogul — does it” defense, if it comes that.
All that said, if Trump is determined to fire him (and cares about it more than the prospects of tax reform), sooner probably makes more political sense than later — it will be an incredible shock to the system and the more time it has to absorb it, the better.