Does God require any other sacrifice?
No. Christ's offering redeems us from all sin, so that no other satisfaction is required.
Catechism for the Global Methodist Church, question 58
12And now, Israel, what does the LORD your God ask from you except to fear the LORD your God, to walk in all his ways, and to love him, and to serve the LORD your God with all your heart and with all your soul? 13You are to keep the commandment of the LORD and his statutes that I am commanding you today, for your good.
21But now the righteousness of God has been revealed apart from law, being attested by the law and the prophets. 22This righteousness of God is through faith in Jesus Christ to all who believe, for there is no distinction. 23For all have sinned and fall short of the glory of God. 24They are justified freely by his grace through the redemption that is in Christ Jesus. 25God presented him as a mercy seat through faith in his blood, in order to demonstrate his righteousness by passing over the sins previously committed. 26This was done in the forbearance of God to demonstrate his righteousness at the present time, so that he might be both just and the justifier of the one who has faith in Jesus.
5For there is one God, and one mediator between God and men, the man Christ Jesus. 6Who gave himself as a ransom for all: the testimony at the proper times.
12But when he had offered for all time a single sacrifice for sins, he sat down at the right hand of God. 13He is waiting from then on until his enemies are made a footstool for his feet. 14For by a single offering he has perfected for all time those who are being sanctified.
This question closes the catechism's teaching on the cross before the Wesleyan way of salvation moves on to justification. The answer is drawn almost word for word from Article VIII of the Confession of Faith, Reconciliation Through Christ, which confesses that the offering Christ freely made on the cross is "the perfect and sufficient sacrifice for the sins of the whole world, redeeming man from all sin, so that no other satisfaction is required." Article XX of the Articles of Religion teaches the same doctrine under the title Of the One Oblation of Christ, Finished upon the Cross.
The word satisfaction is the old term for what is owed to God on account of sin, and the catechism says that Christ has paid it in full. Hebrews puts the finished quality of that payment in the tense of its verbs: "when he had offered for all time a single sacrifice for sins, he sat down at the right hand of God" (Hebrews 10:12). "For by a single offering he has perfected for all time those who are being sanctified" (Hebrews 10:14). God "presented him as a mercy seat through faith in his blood," so that God "might be both just and the justifier of the one who has faith in Jesus" (Romans 3:25-26). No one else could add to that work, because (1 Timothy 2:5-6)
Deuteronomy 10 is on the list for a different reason. Moses asks what the Lord requires of Israel and answers with fear, love, and service "with all your heart and with all your soul" (Deuteronomy 10:12). That is a description of what God asks from a people he has already redeemed; it says nothing about a payment for sin.
For a Methodist this teaching settles the conscience. Nothing done in church or in private adds to the price Christ paid. The Lord's Supper is received as a remembrance of that one sacrifice and as a means of grace, never as a repetition of it, which is why Article XX of the Articles of Religion rejects any teaching that Christ is offered again. Good works and fasting come after the cross, as fruit; the satisfaction for sin was finished when Christ sat down (Hebrews 10:12).