From Talking Points Memo, Judd Gregg, a Republican Senator is supposedly wanted as the Commerce Secretary, perceptively bipartisan, but with the appearance of political cynicism from the Federal character of the US system as the state Governor appoints the replacement Senator:

Of course, New Hampshire has a Democratic governor who would presumably send another Democrat to Washington, becoming the Democrats' sixtieth senator.

Which is a filibuster proof majority. However, Australia has some history with Premiers breaking convention and appointing Senators of their own party; leading to a constitutional crisis:

Following the appointment of the government's Senate leader and Attorney-General, Lionel Murphy, to the High Court bench in February 1975, the Liberal government of New South Wales, under Premier Tom Lewis, refused to follow convention and appoint a Labor replacement for Murphy in the Senate. The independent Mayor of Albury, Cleaver Bunton, was appointed instead.

Following the death of Queensland Labor Senator Bert Milliner, the Country Party Premier of Queensland, Joh Bjelke-Petersen, also refused to appoint a Labor replacement, opting instead to appoint Albert Patrick Field.

The change in Senate numbers was what led to the blocking of supply. In 1977 a constitutional amendment was passed in the Australian constitution that required the state government to fill a Senate seat with a member of the same party. This added the word 'political party' to the Australian constitution, losing its party agnosticity.

While the appointment of Gregg to the executive and the replacement of an elected Republican Senator with a Democrat won't precipitate a constitutional crisis in the US, the use of that power by the states for cynical political purposes may lead to a result that is politically as devastating to constitutional government as the dismissal was.

More Reading on Judd gregg

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