What arguments were made against William Carey's missions efforts?

This is unlikely to be the ideal answer you seek, but after 2 years and 9 months it is the only one offered to date.

I have looked through two reputable books on the subject, ones that deal with the events of William Carey's mission to India, seeking to find any information about the difficulties he experienced in starting his overseas mission. However, both those books were written in the late 1970s and not by those who opposed him in the 1790s. From what is written, it appears that there were two main difficulties he encountered. One was theological, the other was more to do with church politics and church governance.

The theological issue is what you seek to explore, so here is a relevant quotation from Stephen Neil's A History of Christian Missions pages 261-2 (The Pelican History of the Church, volume six):

Carey, who had turned Baptist at the age of eighteen, combined for a time the offices of village pastor and village cobbler... One of the first fruits of Carey's zeal for study was his treatise An Enquiry into the Obligation of Christians to use Means for the Conversion of the Heathen (1792). The title indicates one of the difficulties with which Carey and those like-minded with him had to contend; the hyper-Calvinism of the day had convinced many that the conversion of the heathen would be the Lord's own work in his own time, and that nothing could be done by men to hasten it. Carey's answer is a patient, methodical survey of the world and of the whole history of Christian efforts to bring the Gospel to it. His mind is entirely free from the eschatological speculations of the pietists... The appeal of this pamphlet was reinforced by Carey's sermon to a group of Baptist ministers at Nottingham on 31 May 1792; starting from the text Isaiah 54:2-3, 'Lengthen thy cords, and strengthen thy stakes', he laid down his two great principles of action: 'Attempt great things for God; expect great things from God.' Four months later the Baptist Missionary Society was formed. In June 1793 Carey left for India with his family as the Society's first missionary.

Tantalising clues are given with mention of hyper-Calvinistic views that men could not hasten the work of God in conversion, and of pietistsic eschatological speculations.

The principles of pietism are the demand for personal conversion and for holiness, close fellowship to the Society, and responsibility for witness. The expectation that the return of Christ, which cannot be much longer delayed, will be preceded by a great outpouring of the Spirit of God on Jews and heathens led by a natural gradation of thought to a sense of responsibility for ‘foreign’ missions. (Ibid p 228)

Hyper-Calvinism has long been a problem in Protestantism, giving Calvinism a bad name. Various biblical arguments have been proposed, to imply that those who are saved by the grace of God have to leave the work of the conversion of others to the Holy Spirit; they simply worship God faithfully, 'fence' the Lord's table so that no 'unworthy' person partakes, and teach from the scriptures. Those who are not saved who hear the teaching might be saved, by the grace of God. Now, when we look at the arguments for the hyper-Calvinists, an interesting point arises. The same scriptures that the Calvinists and the hyper-Calvinists use to uphold the predestining sovereignty of God in election are interpreted differently by both groups. Different interpretations produce different theological conclusions, which either encourage reaching out to the unsaved with the gospel, or discourage that. Great lumps of quotations could be given to show where the groups differ, but no scriptures would be involved. The books that deal with the differing stances are all about interpretations of scriptures, not the scriptures themselves.

There is an excellent book dealing with this, The Story of Christian Theology, pages 455 following, by Roger E. Olson (IVP 1999). Although it does not deal with missions, it shows the theological issues between Calvinism and hyper-Calvinism. But it does not thrash out scriptures. It traces the development of theological ideas based on different interpretations and emphases of scriptures. One way of expressing this could be that such debates about predestination and the sovereignty of God showed the limits of where ideas and words could take Christians, theoretically. Carey was not concerned about that; he wanted to take the Gospel to lost souls, trusting in God to do great things as Christians obediently did everything they could to fulfil Christ’s commission in Matthew 28:19-20.

This does not exactly answer your question but given how missions have grown and shown great success, that may indicate why there seems to be such a dearth of information on the objections of the hyper-Calvinists in centuries past. Had their arguments been sound, we could have expected dismal failure on the part of the missionary societies, with hyper-Calvinistic arguments being upheld to this day as proof that they were right. It appears that they were wrong.