Your question brings up the entire Ebers-Moll model and correctly points out that both forward and reverse \$\beta\$ values would be needed.
I discuss the saturation case in detail here, with respect to the injection version of the Ebers-Moll model. When saturated, the full model is actually required and you typically need to know both the forward and reverse \$\beta\$ values to get an accurate result. (Not always, as your circuit will illustrate in an example at the bottom of my answer here.)
However, your case appears to be in active mode. That fact allows the use of only \$\frac12\$ of the Ebers-Moll model. And as you point out in your question, while technically both forward and reverse \$\beta\$ values are still required, neither are necessary as evidenced in your equation.
(I also discuss a couple of versions of seeing the bipolar model here. Perhaps that may also be helpful. There are three versions, but I only discuss two of them here. If you want to see all three versions, then look here.)
The injection version would have:
VT = 25.8649606e-3 # measured VTISAT = 5e-17 # model ISdef A(b): return 1/(1+1/b) # alpha factordef IC(vbe,vce,vt,isat,br): return isat*exp(vbe/vt)*(1-exp(-vce/vt)/A(br))def IE(vbe,vce,vt,isat,bf): return isat*exp(vbe/vt)*(exp(-vce/vt)-1/A(bf))def IB(vbe,vce,vt,isat,bf,br): return isat*exp(vbe/vt)*(1/bf+exp(-vce/vt)/br)But a quick glance at the equation for IC above, assuming active mode, just provides for the equation you noted in your question: \$I_{_\text{C}}\approx 50\:\text{aA}\cdot\exp\left(\frac{800\:\text{mV}}{V_T}\right)\approx 1.3541\:\text{mA}\$.
Assuming no Early Effect, then the collector voltage will be \$2.5\:\text{V}-1.3541\:\text{V}=1.1459\:\text{V}\$, confirming a simpler active mode requiring only half the Ebers-Moll model.
If you are serious about exploring the fuller Ebers-Moll model, then this requires saturation mode. You can achieve that by increasing the collector resistor's value to \$1.8\:\text{k}\Omega\$. In this case, you really will need to know the reverse \$\beta\$ value -- if not the forward \$\beta\$ value:
def FindVCE(vbe,vce,vcc,rc,isat,betaR,vt): vce0=vce while True: vce1=vcc-IC(vbe,vce0,vt,isat,betaR)*rc if abs(vce0-vce1)/vce0 < 0.1: break vce0=(vce1+vce0)/2 return vce1FindVCE(800e-3,0.5,2.5,1.8e3,5e-17,3,VT)0.109600415871553Which definitely says the bipolar is saturated now, with \$V_{_\text{CE}}\approx 110\:\text{mV}\$.
Let's test this result with LTspice:
Looks as though Ebers-Moll accurately predicted the result found using a Spice program (and a more sophisticated model, but where I reduced it back to a more ideal version closer to Ebers-Moll.)
It doesn't get much better.