Is compatibilism compatible with the doctrine of the Catholic Church? My usual reasoning was that it was not on the basis that:


Determinism would entail predestination, which is rejected by the Catholic Church;
The alternative - quantum mind - is trickier, but:


To my intuition, true randomness entailed by quantum mechanics make it impossible for people to be responsible for their actions, which is necessary for the doctrine of sin;
Quantum brain is dubious at best anyway.



However, it should be noted that in spite of scientific arguments against incompatibilist free will, I can't see the Church vehemently opposing these arguments, hinting that the Church might not view the rejection of incompatibilist free will as a threat to its doctrine; which hints that compatibilism is, in fact, compatible with the doctrine of the Catholic Church.

What is the state of affairs?
The Catholic Church Rejects Determinism and Compatibilism

My argument for the Catholic rejection of Compatibilism is this: Compatibilism entails the rejection of Libertarian Free Will*, but Catholicism affirms Libertarian Free Will and rejects determinism, therefore Catholicism rejects Compatibilism.  Here is some support for the second premise, the Catholic affirmation of the commonsensical understanding of free will:


  Freedom is the power, rooted in reason and will, to act or not to act,
  to do this or that, and so to perform deliberate actions on one's own
  responsibility. By free will one shapes one's own life. (Catechism of
  the Catholic Church, 1731)





  Man has free-will: otherwise counsels, exhortations, commands,
  prohibitions, rewards, and punishments would be in vain. (Thomas Aquinas, 
  Summa Theologiae, Ia, Q 83, A 1)





  Some have held that the human will is necessarily moved to choose
  things.  But they did not hold that the will is coerced, since only
  something from an external source, not everything necessary, is
  coerced.  And so also some necessary movements are natural but not
  coerced.  For what is coerced is as contrary to what is natural as to
  what is voluntary, since the source of both the natural and the
  voluntary is internal, and the source of what is coerced is external.
  
  But this opinion is heretical.  For it takes away the reason for merit
  and demerit in human acts, as it does not seem meritorious or
  demeritorious for persons to do necessarily what they could not avoid
  doing.
  
  It is also to be counted among the oddest philosophical opinions,
  since it is not only contrary to faith but also subverts all the
  principles of moral philosophy.  For if nothing is within our power,
  and we are necessarily moved to will things, deliberation,
  exhortation, precept, punishment, and praise and blame, of which moral
  philosophy consists, are destroyed.  And we call like opinions that
  destroy the foundations of parts of philosophy odd, as, for example,
  the position that nothing is moving, which destroys the foundations of
  natural science.  And some individuals were indeed led to hold such
  positions partly because of impudence, partly because of sophistical
  arguments that they could not refute, as the
  Metaphysics15 says.
  
  -Thomas Aquinas, Quaestiones Disputatae de Malo, Q 6, A 1 (Tr. Richard Regan)





  Man was not weakened to the point where the moral life would be
  impossible for him: he still retained free will, understood as freedom
  from necessity: not only from aviolence or
  bconstraint, but also with respect to cthat
  necessity which was voluntary only in the original sin as its cause
  (Adam), 1939, 1941, 1952, a1966f., b2003,
  c2301; free will is useful not only with respect to sin,
  1927-1930, 1965, 2438-2440; the value of free will is defended against the affirmations: [It was totally adestroyed,
  bit is an empty concept, cit is an invention
  of Satan], a331, a336, a339,
  b1486, abc1555, 6245f.; cf. L 1b (contingent
  freedom, obliged to the good); L 1f (moral act).
  
  -Denzinger Enchiridion, Systematic Index D 2bc





  The consciousness of freedom and human dignity as important
  characteristics of our time, 4750; cf. C 4kc (contemporary changes);
  the value of freedom promises an advance of man toward perfection,
  4505; cf. C 4ie (progress).
  
  Man also possesses freedom in the state of fallen nature: D 2bc
  (effect of original sin); if there were a fated necessity, it would
  suppress the responsible character of human actions such as reward and
  punishment, 283.
  
  Because of free will, man is autonomous and can act freely, 4752;
  freedom confers on him the dignity of having power over his actions,
  3245, 4752; his dignity demands that he act by a free and conscious
  choice, that is, personally, 4317; only in freedom can man direct
  himself toward the good, 4317; he is obliged to observe God's
  commandments through free will, 227, 245; freedom as an intrinsic
  attribute of the human person, 4765; vocation of man to full freedom,
  (4752), 4815; even if it is limited by circumstances, freedom is not
  entirely suppressed, 4754; cf. L 1b (contingent freedom, obliged to
  the good); L 1f (moral act).
  
  -Denzinger Enchiridion, Systematic Index C 4fc




It is perhaps worth noting that the Catholic position on issues such as these is inevitably nuanced and robust.  For example, philosophically and theologically it would be more common to say--rather than what you said in your question--, "Predestination would entail Determinism, which is rejected by the Catholic Church."  And yet the Catholic Church does not deny predestination, for She does not believe predestination to entail determinism.

*Libertarian Free Will can be a rabbit hole in philosophical discourse, but I am using it in a very simple and common way.  By the term I mean that commonsensical free will exists, humans have the ability to do otherwise, and determinism is false.  The ability to do otherwise means that if I chose A then I could have chosen B.

(Quoting from Denzinger's Systematic Index carries with it the danger of confusion, and yet I think it is warranted due to its comprehensiveness.  Each of the references from the SI points to a different Catholic dogma or doctrine promulgated at some point in the past 2,000 years.)