I take it as undisputed that: if you tell someone the truth, you've been sincere or genuine with them.
What is the "offer of the Gospel?" Put simply: Romans 10:9 if you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved.
The argument is frequently stated that "If you're a 5-point Calvinist then you believe in limited atonement, thus you cannot make a sincere offer of the Gospel."
Before an answer is given, let's note something about conditionals. A conditional statement is, "an “if p, then q” compound statement (ex. If I throw this ball into the air, it will come down); p is called the antecedent, and q is the consequent. A conditional asserts that if its antecedent is true, its consequent is also true; any conditional with a true antecedent and a false consequent must be false. For any other combination of true and false antecedents and consequents, the conditional statement is true."
So a conditional statement, taken as a whole statement, has a true truth value just in case the consequent is not false while the antecedent is true. To use the above example, if I say If I throw this ball into the air, it will come down, then I have made a true statement, irregardless of whether or not I in fact throw the ball into the air or not.
Therefore, going back to the issue at hand, since it is impossible for someone who believes to not be saved, then when we tell anyone that "if you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved," then we have told that person the truth. As I said above, I take it as axiomatic that "if you tell someone the truth, you have been sincere with them." Thus we can make the argument:
1) If you tell someone the truth, you have been sincere with them.
2) When a Calvinist makes a Gospel offer to anyone he is telling that person the truth.
3) Therefore, when a Calvinist makes a Gospel offer to anyone, he has been sincere with them.
Thus Calvinism has no problem with "the sincere offer of the Gospel."