I think your analysis is fairly correct, I'll try to explain why it makes sense, and what falls a bit short.
When you have springs in series, the force applied acts on each spring. For the case where you have two identical springs ($k_1 = k_2$, $x_1 = x_2$) in series, you can see that to get the displacement $x_1 + x_2$, $F_1 = F_2 = F_3$. This is because since the force acts on each spring in series, to compress both by that amount, you can just apply that same force.
When the two springs are different, it doesn't work out that nicely. Your analysis shows that the force $F_3$ will always in between the two $F_1$ and $F_2$ which makes sense, at least mathematically.
The issue is with the wording fully compressed. When dealing with springs that cannot compress beyond their maximum value (i.e. $x_1$ is as far as spring 1 will compress), then the simple math doesn't actually work.
In such situations, the force would always be the force required to compress the stiffest spring to it's maximum value. That is because applying more force to a fully compressed spring does not compress it any further.
In the case where $k_2 > k_1$, for example, when you apply $F_1$, the spring with $k_1$ compresses fully, and the other spring only compresses some. If you add more force, spring 1 doesn't compress any more; but the equation you were using doesn't account for that. Instead, you are analyzing the situation where $F_3$ is applied ($F_1 < F_3 < F_2$), and the math would say that spring 1 is compressed more than $x_1$ when $F_3$ is applied (and spring 2 is compressed less than $x_2$; but the total compression is still $x_1 + x_2$) but if $x_1$ is the maximum compression, that math obviously doesn't represent reality.
Once you recognize that spring 1 can't actually compress more than $x_1$, and spring 2 can't compress more than $x_2$ you should be able to see that therefore, to displace both springs in series as much as $x_1 + x_2$, you actually need to apply the force required for the stiffer spring; not the in-between force $F_3$.
Sorry this got long winded, I just really like springs.