How do Christians argue about Jewish objections against Jesus being the Messiah

There is no unique argument from the Catholic Church about those objections, but being myself a Catholic I will offer mine. Since IMV this argument has nothing specifically Catholic about it, I will speak just as a Christian. Sure enough, many Christians may disagree with this answer, particularly with the work on the epistemological foundation.

Core of the argument

Christians do not argue about those objections by answering each of them with the Christian interpretation of the corresponding OT passage, as that would be missing entirely the basis of our faith in Jesus as not only the Messiah but also the consubstantial Son of God Who has assumed a human nature. Because we do not hold Jesus to be the Messiah or the Son of God on the basis of believing that our interpretation of the OT is so better than the Rabbinic one, but on the basis of believing the FACT of Jesus' resurrection. Facts trump interpretations, or as Pope Francis says, reality is superior to ideas. If Jesus in fact rose from the dead, then what He claimed about Himself is true and any differing OT interpretation must be discarded, period.

Let me hammer the point again. The logic of the Christian argument is not "The Christian interpretation of the OT is really good, therefore the events in the Gospels must be factual" but the other way round: "Jesus' resurrection is factual, therefore Jesus' words and deeds according to the Gospels are true, i.e. both historically factual and truthful, and therefore the Christian interpretation of the OT is correct."

BTW, the answer in JSE that you linked to not only does not even address the issue of whether Jesus' resurrection was factual, but also gets the issue altogether wrong, as it says "Now, many missionaries tell me that Jesus will rise from the dead and do all the requisite things that Messiahs are supposed to do that he didn't get to." Were those missionaries drunk or speaking in tongues? The point is that Jesus ROSE from the dead, not that He "will rise"!

Epistemological foundation

Now, since the answer so far might be countered with the objection that Jesus' resurrection might just be an instance of the "sign or wonder" supporting the claims of the false prophet predicted in Deut 13:1-5, we must start from a solid philosophical foundation in order to be able to discern correctly whether a sign or wonder is intended to lead people to the one true God or away from Him. For that purpose, I define a Revelatory Epistemological Position as the set of criteria that a person holds to be appropriate for determining whether a purported medium of divine Revelation is actually so. (Or in the case of a particular sign or wonder, whether it certifies that the associated revelation comes from God or it does not.) Thus, the logic in the above statement "Jesus' resurrection is factual, therefore Jesus' words and deeds according to the Gospels are true, i.e. both historically factual and truthful," presupposes a specific Revelatory Epistemological Position which consists of the following:

A first postulate based on infinite divine goodness:

P1. God will perform a miracle that requires the exercise of God's own exclusive power (as opposed to allowing an angel to exercise its own natural power) - in scholastic terms, a miracle which can happen only if God performs it directly as its only cause, not as its primary cause cooperating with a natural secondary cause, whether material or angelic, - hereafter called "a miracle which only God can perform", only to lead people to truth and good.

plus two corollaries of P1:

C1. The possible instances of the "sign or wonder" supporting the claims of the false prophet predicted in Deut 13:1-5 do not include miracles which only God can perform, but are limited to the works that an evil angel can perform by its own natural power when allowed by God.

C2. If God performs a miracle which only He can perform through the intervention of a prophet P, God will be certifying in front of everyone that all that P has said and done as prophet so far has God's "seal of approval" and therefore is true and in accordance with God's will, respectively.

plus a second postulate based on metaphysics:

P2. The resurrection of a human being is a miracle which only God can perform, regardless of whether God performs it through the intervention of a human being, such as prophet Elisha (2 Ki 4:32-35).

plus two further corollaries from P2 and C1 and C2, respectively:

C3. The possible instances of the "sign or wonder" supporting the claims of the false prophet predicted in Deut 13:1-5 do not include the resurrection of the false prophet himself to a glorious state.

C4. If God resurrects a prophet P to a glorious state, God will be certifying in front of everyone that all that P has said and done as prophet up to his death has God's "seal of approval" and therefore is true and in accordance with God's will, respectively.

plus a third postulate based on infinite divine wisdom and goodness:

P3. If God confers his "seal of approval" to the words and deeds of a prophet P by resurrecting him to a glorious state, He will also provide the means to ensure that the words and deeds of said prophet can be known with accuracy and certainty along time and across space. Because it would be against divine wisdom and goodness for God to give us certainty that whatever P said and did was true and good while not giving us certainty about what P actually said and did!

Starting from the above revelatory epistemological position and obtaining, through rational examination of historical facts, a certainty about the historicity of Jesus' resurrection, one arrives at the certainty of the factuality and truth of Jesus' words according to the NT.