There is a problem for their position: it is not the position of their prophet, assuming that Surah 62:5 is itself not corrupt.
Surah 62:5 The similitude of those who were charged with the (obligations of the) Mosaic Law, but who subsequently failed in those (obligations), is that of a donkey which carries huge tomes (but understands them not). Evil is the similitude of people who falsify the Signs of Allah: and Allah guides not people who do wrong.
Notice that the Surah compares the Jews to Donkeys carrying "huge tomes" that they do not understand. Notice that the Surah alleges that the Jews "falsify the Signs of Allah," but the most natural reading of the Surah (at least in this translation) is that they falsify not through destroying the text, but through misunderstanding or misinterpreting the text.
Notice that is says "huge tomes" not "no tomes" or "tattered tomes."
And, in another translation, we see the same concept:
62.5 The likeness of those who were charged with the Taurat, then they did not observe it, is as the likeness of the ass bearing books, evil is the likeness of the people who reject the communications of Allah; and Allah does not guide the unjust people.
Notice how the text parallels "books" and "communications of Allah." Notice also that in this translation they are described as rejecting them - presumably because like donkeys they don't understand what is in them - perhaps simply because "Allah does not guide" them.
And it is not the only place in the Koran that such an idea can be found. Recall that it is recorded:
[2.285] The apostle believes in what has been revealed to him from his Lord, and (so do) the believers; they all believe in Allah and His angels and His books and His apostles; We make no difference between any of His apostles; and they say: We hear and obey, our Lord! Thy forgiveness (do we crave), and to Thee is the eventual course.
And in another translation:
Surah 2:285 The Apostle believeth in what hath been revealed to him from his Lord, as do the men of faith. Each one (of them) believeth in Allah, His angels, His books, and His apostles. "We make no distinction (they say) between one and another of His apostles." And they say: "We hear, and we obey: (We seek) Thy forgiveness, our Lord, and to Thee is the end of all journeys."
Notice that it is says "books," not "book." What's even more interesting is that we are told that the Surahs were themselves orally proclaimed at first - which would suggest that the "books" do not even include the Koran. There is no need, however, to go that far. The fact that the plural form is used is enough to establish the fact that Mohamed did not hold to the idea that the books were corrupted, for if they were corrupted why would he teach that he believes in them?
And we need not rest on only those two places, for there is at least one more:
[20.133] And they say: Why does he not bring to us a sign from his Lord? Has not there come to them a clear evidence of what is in the previous books?
And again, in another translation:
Surah 20:133 They say: "Why does he not bring us a sign from his Lord?" Has not a Clear Sign come to them of all that was in the former Books of revelation?
Note how Mohamed here actually states that there is "clear evidence" or a "clear sign" as to what the content of the "former books" or "previous books." These are not books hopelessly lost in obscurity in the mind of the author of this Surah.
The problem here is that Mohamed simply was not well informed. He did not know the content of the books himself, for we have no reason to believe he was ever provided with an Arabic Bible, and he could not read the Greek or Hebrew original languages.
If you are a Muslim, doesn't this sort of thing cause you a bit of concern? Aren't you bothered by the fact that your prophet accepted the books and "believed in them," while you must not? For the books declare that Jesus is the Christ, the Son of the Living God - God incarnate - the sacrifice to satisfy the justice of God and bring forgiveness to all the people of God.