According to literal dispensationalism, how does the short time of Rev. 12:12 last for 2,000 years or more?

From the Dispensationalist (at least the pre-tribulation variety), your understanding of Revelation 12 is flawed.

Dispensationalists understand the Child to be both Christ and His Church. You allude to the same in mentioning the persecution of early Christian by the Sanhedrin and Nero. But when does this assault by the dragon end? Revelation 12:5-6 is key.

She bore a male Child who was to rule all nations with a rod of iron. And her Child was caught up to God and His throne. Then the woman fled into the wilderness, where she has a place prepared by God, that they should feed her there one thousand two hundred and sixty days. (Revelation 12:5-6, NKJV)

In many circles, 'her Child was caught up to God and His throne' refers to the Ascension. But this is not the Dispensationalist understanding. Remember, the Child represents both Christ and the Church. Therefore, 'caught up' can not refer to the Ascension. Rather, it is understood to be the Rapture where Christ returns (at least partially) to collect His Church. Compare with Paul's description of the Rapture:

For the Lord Himself will descend from heaven with a shout, with the voice of an archangel, and with the trumpet of God. And the dead in Christ will rise first. Then we who are alive and remain shall be caught up together with them in the clouds to meet the Lord in the air. And thus we shall always be with the Lord. (1 Thessalonians 4:16-17, NKJV)

Note in particular to the phrase 'caught up'. This is an important connection in Dispensational thought and is believed to cleanly separate the Rapture in Revelation 12:5 from the Ascension.

J N Darby writes in his commentary on Revelation 12:

But Satan, as the open infidel enemy of God and God's power in Christ, sought to devour the child as soon as born, who was to have the rule of the earth from God. But the child, Christ, and the assembly with Christ, is caught away to God and His throne does not receive the power yet, but is placed in the very source of it from which it flows. ... Now, if the man-child in heaven be, as we have considered it, Christ and the raptured saints, the voice would be that of those already there, and all self evident: the raptured saints associated with Him celebrate the casting down of the accuser and the deliverance of those who belonged to heaven

William Kelly writes much the same:

On this mystical principle then the rapture of the Man child to God and His throne involves the rapture of the saints in itself. The explanation why it could be thus introduced here depends on the truth that Christ and the church are one, and have the common destiny of ruling the nations with a rod of iron. Inasmuch as He went up to heaven, so also the church is to be caught up.

H A Ironside concurs:

We have seen that the man-child symbolizes both Head and body-the complete Christ. Therefore, as in other prophecies, the entire present dispensation is passed over in silence. The church is represented in its Head, caught up with Christ.

So, in the Dispensationalist view, Revelation 12:5 has not been fully fulfilled. The last 2000 or so years are contained in that verse. Ironside calls this out: "Therefore, as in other prophecies, the entire present dispensation is passed over in silence." This is a common belief in Dispensationalism -- that the Church Age was hidden from the prophets. The present dispensation, the Church Age, will end with the Rapture. It is not until then that Revelation 12:6 will be fulfilled with the woman (Israel/Jews) fleeing into the wilderness.

So, now we can address the original question:

How do they reconcile a period of 2,000 or more years since the birth of Christ and the birth of the church in the first century A.D. to the "short time" of Rev. 12:12 that the devil was permitted to try to defeat Christ?

The 'short time' refers not to the last ~2000 years, but to the Tribulation that follows the Rapture. According to Dispensationalism, the Tribulation will last for seven years -- a relatively short time.