Epistemic rationality is honestly positioning one’s degree of belief in a proposition to the degree that the balance of evidence for that proposition warrants. Epistemic rationality belongs to the individual epistemic agent.
Burden of proof is more of a social concept. If someone wants to convince members of a community a proposition that is incongruent with the current beliefs of that community, then, to accomplish that goal, he or she needs to take on the onus of presenting the arguments/evidence that will accomplish that.
Note that this assumes that the community values arguments and evidence. What could it mean to have a burden of proof where proof (read “convincing evidence” here), is not valued among epistemic agents. If the epistemic agents within a community hold that beliefs need not map to the balance of evidence, then the notion of burden of proof will be of no value to them.
So, what is logically prior? Epistemic rationality is prior to burden of proof. If a community thinks, to any degree, that they need not map their degree of belief in X to the degree that X is warranted by the confirming/disconfirming evidence, then they only foolishly inform those who would attempt to provide arguments/evidence for a position not their irrational own that they must take on the burden of proof.
Burden of proof is like the status of offense or defense in a football game. If you don’t respect the rules of football, or want to play football under the rules of tennis, you don’t have any right to suggest the other team must now play offense. Until you demonstrate a willingness to play by the rules of football, you don’t have any right to suggest who takes possession of the ball.
Rationality comes first. Only after you have committed to mapping your degree of belief to the degree of the confirming/disconfirming evidence can you suggest the other side has the burden of providing evidence/argumentation that might change your mind.