Two of Tony Abbott’s long-promised industrial relations crackdowns appear headed for Senate defeat with the Palmer United party and crossbench senator Ricky Muir set to join Labor and the Greens in rejecting them.

The prime minister promised before the election to re-establish the Australian Building and Construction Commission as a “tough cop on the beat” for the building industry and to set up a new registered organisations commission to monitor the conduct of unions and business groups.

Labor and the Greens oppose both bills and now a spokesman for the Palmer United party has confirmed both PUP senators intend to vote against them. A spokesman for Motoring Enthusiast senator Muir said he was also “very likely” to vote against them.

Both bills were set to be debated this week but have now been deferred by the government as it seeks more time to lobby and win the crossbench votes.

The looming Senate defeat comes as the government is seeking to recover from Monday’s damaging leadership spill motion and as Abbott said Senate obstructionism had been the only mistake in last year’s budget.

He said the only thing the government got wrong with its 2014 budget was that it had “failed to get legislation through ... a Senate controlled by our political enemies” and that the only promises he had actually broken were spending cuts to foreign aid and the ABC.

The government has given mixed messages about whether it remains committed to key budget measures stalled in the Senate, including higher education reforms and the Medicare co-payment, which has already been twice revised.

The tougher industrial laws were part of the government’s election pitch, and are often touted as the answer to scandals such as the wrongdoing at the Health Services Union.

The building and construction industry (improving productivity) bill was introduced almost as soon as the government won office in 2013, and the fair work (registered organisations) amendment bill – described by Abbott as “very significant legislation” – was introduced in the middle of last year.

Labor argues legislative changes it made in government already strengthened the Fair Work Commission’s investigative powers and penalties were increased.

In a submission to a Senate committee early last year, the workplace relations minister, Eric Abetz, said: “The government considers the fair work (registered organisations) amendment bill 2013 as a high priority piece of legislation ... This policy has been well ventilated for some time and the government has a very clear mandate to implement it as a matter of extreme urgency.”