But, you need a much, much larger TPS, the mass of which would negate any advantage to the no upper stage option. A heavy upper stage with sufficient TPS would mean a burnout velocity for the core that is only slightly faster than the standard F9.
Do you have numbers to back this up? Some back-of-the-envelope calculations: Apollo-CM heat shield was around 1/3 of the landed weight. A modern lightweight material like PICA-X weighs about half of that if I've understood things right. Assume 20 tonnes landed weight for the first stage. That gives you, what, 4 tonnes worth of heat shield? I don't see how that can change the equation.
A few things: Apollo's heatshield was about twice as thick as it strictly had to be, and it also was returning from the Moon (not just LEO). With modern materials, you could probably cut that number down to just 5% of the landed weight if you were careful.
Also, the first-stage isn't going to be NEARLY that fast at staging. It's just going to be going a few times the speed of sound. I think you're thinking of the UPPER stage, which for the expendable Falcon 9 v1.1 may weigh about 5mT burnout, not 20mT....
Oh, wait, you were talking about using Falcon 9 v1.1 first stage as a SSTO? Horrible idea. It should be possible if you're talking about expendable (though it would have a horrible mass fraction compared to a similar take-off mass TSTO expendable, and it'd be strictly for LEO... it'd be more expensive per kg to LEO than a TSTO), but reusably, it's DEFINITELY not feasible and maybe not even possible.
EDIT: Nevermind, I see you were talking about the center core of the Falcon Heavy. It might be possible to do what you suggest, might even be worth it.