On the Catholic view of the atonement

When I medidate on the Passion of Christ, I end up falling into the same doubts about the atonement, which I want to ask. The Catholic Church denies the penal substitutionary atonement, i.e. the notion Jesus presented himself before the Father so that he may be punished on our behalf. Instead, to atone for an offense is to offer to the Offended something that He loves equally or even more than He hated the offense, and so, because sin is an offense to God, the Church teaches that the sacrifice of Christ to the Father is this offering on our behalf, which, in virtue of Christ being the Son of God, is more pleasing to the Father than the whole collective of sin of humankind. Furthermore, the suffering, crucifixion and death of our Lord were meritorious of all grace to us, this making sense of the seven sacraments, the sacramentals and the spiritual authority of binding and losing of the Church. **My question:** I admit that my doubts, and thus my question, is half driven by emotions. My doubt is this: "Sacrifice" in more general terms can just mean offering for the sake of the one to whom we offer, e.g. I can offer to God my time in prayer and meditation, or my intellect in faith, or my will in obedience, for the sake that He, being God, is the ultimate object of my desire. Then why is it that Christ's sacrifice needs to be in the sense of Christ giving His life to suffer and die on the cross, instead of just an offering of Himself in a less bloodshed way? I know that God could save us in any other way, for He is omnipotent, and that He chose the cross because He thought of it as the fittest way. However, on this I came to another facet of my doubt, i.e. when Christ was on the Gethsemane He said: > Father, if it is possible, let this cup pass from me, but not as I will but as You will. implying that His death on the cross was of the will of the Father. So, how can the Father is pleased in the sacrifice of His Son that He wills? For, when I imagine my son sacrificing for the sake of another, I truly understand and can't help but to love my son for it, but not as my son sacrificing himself for the sake of my will. Again, this is half driven by feelings, but these often get in the way of my spiritual life so I want a way to resolve this. I appreciate any comment, and God bless.