Is an X ml dose of a drug at 50% bioavailability roughly equivalent to a 2X ml dose at 25% bioavailability?

For the sake of simplicity, assume:

• The two doses are the same drug metabolized in the exact same way, and the only difference is the amount that reaches the bloodstream (and thereafter the brain).

• This drug is only meant to influence the brain, so differing concentrations in various organs don't matter.

• This drug is short-term and nothing lingers in the body after it has been excreted or broken down.

My understanding is that bioavailability means the proportion of drug that reaches the bloodstream. So, all things being equal, does that mean an administration of a drug will be twice as potent if the bioavailability is doubled?

Bioavailability is initially taught, and sometimes reported as a proportion, $F$, the concentration in drug in plasma under a test condition (typically oral administration) over the concentration of of the same dose of the drug under a reference condition (typically IV administration).
In the simplest case, under zero order kinetics (the rates of all reactions that cause a change in concentration of the drug are independent of that drug's concentration), you can substitue $F\cdot D$, for a dose, where $F$ is the bioavailability and $D$ is the dose (see equation 2-16 in Goodman and Gilman's The Pharmacological Basis of Therapeutics, Chapter 2).
So, for your question, you can write down the state with equal plasma concentrations, $0.5\cdot D_1 = 0.25 \cdot D_2$, which reduces to $2D_1 = D_2$
Potency is a hairy term, defined in many different ways (some with more hand waving than others), but lets say for the purposes of this question that potency is the oral dose $D$ required to produce a measurable effect, $E$, and relative potency is the oral dose of formulation 1, $D_1$ required for a set effect, $E$, over the oral dose of formulation 2, $D_2$ for the same effect (modified from Goodman and Gilman Chapter 3). So, if the effect of your drug is directly proportional to its concentration in plasma ($E = kC$), then yes, the formulation with twice the bioavailability will be twice as potent.
I was going to write a long treatise on why this is not necessarily going to be the case, but since you were so specific about all things being equal, I'll just say this: bioavailability is often not reducible to a single $F$, which is why the FDA requires a lot of information on Bioavailability. This is for good reason, because kinetics are often not zero order, and even once we get to plasma concentration, the concentration term that drives efficacy can be $AUC$, $C_{max}$, time above a threshold, and sometimes even hard to correlate directly to plasma concentration at all. But, in the simplest case, all things being equal, your answer is yes.