How about you wrote down the equations of your so-called "experiment"? Maybe Trooper can help you.
Let one object be at point $$A$$, the other be at point $$B$$, and the AND gate be at point $$G$$. Let $$S$$ be the rest frame of $$G$$, and $$S'$$ be moving with a velocity of $$-.5c$$ with respect to $$S$$. In any frame, until the objects start moving, all three points are co-moving. In $$S$$, they are co-moving with a separation of $$d$$. Length contraction gives us the separation in $$S'$$:
$$d'=d\sqrt{1-.5^2}=\frac{\sqrt{3}d}{2}$$
As soon as object $$A$$ is launched, the gap between it and the gate will start decreasing at a rate $$v'_A-v'_G$$, and object $$A$$ will hit the gate after a time $$T'_A=\frac{v'_A-v'_G}{d'}$$. The same reasoning applies to object $$B$$, except with a minus sign in front of the $$d'$$ in the denominator. Trivially, the gate is moving at $$v'_G=.5c$$ at all times in $$S''$$. From
this helpful website, the post-launch velocities of the objects are given by:
$$\begin{align}
v'_A&=\frac{.6c+.5c}{1+.6*.5}\approx.846c\\
v'_B&=\frac{-.6c+.5c}{1-.6*.5}\approx-.143c
\end{align}$$
These match eram's results. Plugging these into the formulae above:
$$\begin{align}T'_A&=\frac{2(.846c-.5c)}{\sqrt{3}d}\\
&=\frac{2*.346c}{\sqrt{3}d}\\
&=.538\frac{2*-.643c}{-\sqrt{3}d}\\
&=.538\frac{2(-.143c-.5c)}{-\sqrt{3}d}\\
&=.538T'_B\end{align}$$
In other words, after their respective launches, object $$A$$ reaches the gate almost twice as quickly as object $$B$$. Importantly, while I used Lorentz transforms to find the various $$v'$$, an experimentalist could reach the same conclusions just by using radar guns to read the velocities of both objects and the gate.
The gate, meanwhile, will tells us that both objects reached it simultaneously (and since the simultaneous intersection of $$A$$, $$B$$, and $$G$$ is a single space time event, all reference frames will agree that it occurs). The radar measurements tell us that object $$A$$ took less time after its launch to reach the gate than object $$B$$ did, while the gate measurement tells us that the objects reached the gate simultaneously. Applying a bit of logic, the two measurements together show that object $$B$$ launched before object $$A$$.