It has been suggested to me that I should present my discussion comments as an answer.

In principle, to "brave" a fine could imply facing the risk of paying a fine, if the imposition of the fine is not certain. However, in principle the meaning of "brave" as a verb is to face or endure something unpleasant in a brave manner, undaunted, without fear:

"endure or face (unpleasant conditions or behavior) without showing fear"
http://oxforddictionaries.com/definition/brave?region=us&q=b...

"to dare or defy ⇒ "to brave the odds "
to confront with resolution or courage ⇒ "to brave the storm""
http://www.collinsdictionary.com/dictionary/english/brave

"to face or endure with courage"
http://www.merriam-webster.com/dictionary/brave

In this case, the source text leaves open the possibility of risk: that the fine may or may not have been imposed. Without further context, that is what I would have assumed. But thanks to Colin's research we know that Mocenigo knew he would have to pay if he did this, and accepted the obligation fearlessly.

The other source I quoted in the discussion reveals that the fine was imposed by an "ancient Law" forbidding anyone from suggesting making any change to the Doge's palace. The law imposing the fine was not a recent development and did not foresee the circumstances that had arisen. It is not that Venice was determined not to repair the damage that had occurred and to punish anyone who suggested doing so. It is simply the old ancestral Venetian determination to preserve their city and make sure no-one ever altered it in the future, added, perhaps, to a sense of invulnerability; the ancient Venetian legislators were not trying to prevent restoration, merely alteration; they presumably did not foresee that restoration of this kind would ever be necessary: that anyone would successfully attack the city. At any rate, everyone in Venice clearly must have known that Mocenigo would have to pay the fine if he went ahead, which he willingly did. A thousand ducats was a huge sum, but then you had to be fabulously wealthy to be the Doge.

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Note added at 1 day1 hr (2012-01-21 12:30:03 GMT)
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My comments in the last paragraph about the reasons behind this fine may well be incorrect; I based them on the mention of an "ancient law" in the source I quoted previously. However, E. V. Lucas gives a different account of this episode, in which the imposition of a fine on those advocating rebuilding the Palace is attributed to the need to avoid expense at a time of war, especially against Genoa:

"The next Ducal tomb is the imposing one of the illustrious Tommaso Mocenigo (1413-1423) who succeeded Steno and brought really great qualities to his office. Had his counsels been followed the whole history of Venice might have changed, for he was firm against the Republic's land campaigns, holding that she had territory enough and should concentrate on sea power: a sound and sagacious policy which found its principal opponent in Francesco Foscari, Mocenigo's successor, and its justification years later in the calamitous League of Cambray, to which I have referred elsewhere. Mocenigo was not only wise for Venice abroad, but at home too. A fine of a thousand ducats had been fixed as the punishment of anyone who, in those days of expenses connected with so many campaigns, chiefly against the Genoese, dared to mention the rebuilding or beautifying of the Ducal Palace. But Mocenigo was not to be deterred, and rising in his place with his thousand ducat penalty in his hand, he urged with such force upon the Council the necessity of rebuilding that he carried his point, and the lovely building much as we now know it was begun. That was in 1422. In 1423 Mocenigo died, his last words being a warning against the election of Foscari as his successor. But Foscari was elected, and the downfall of Venice dates from that moment."
E. V. Lucas, A Wanderer in Venice (New York: Macmillan, 1914), pp. 260–61.
http://www.gutenberg.org/files/16705/16705-h/16705-h.htm

The interpretation of the phrase remains the same, however; what Mocenigo chose to do made the fine inevitable.