A sine qua non for the US conquest of Iraq was that the bulk of the foreign contracts for the exploitation of its oilfields had been awarded to French and Russian companies, while US-based firms like Bechtel, Amec, and Halliburton (of which Cheney had been CEO) were forbidden by US law to work for Hussein's Iraq.
If we might have had the Russian work going to BP - perhaps along with BP's takeover of Amoco being unsuccessful - then Russia would be much more inclined to take part in the operation if the award of some of those exploitation contracts was accepted as the reward for their involvement. This would be discouraged by the US oil industry lobby and by the UK, which Wikipedia says contributed 45,000 troops to the initial invasion. While that number sounds high to me, it was far and away the biggest contribution to the effort from any other country, and while I can see the Bush administration stripping an Amoco-less BP of contracts to give them to Halliburton/KBR, I can't see the White House doing so to give them to Gazprom.
The bulk of Iraqi oil is located either in the Kurdish north-west or near Basra on the Gulf coast - and the US would be extremely reluctant to permit significant Russian interests in either place. Giving Russia a port on the Persian Gulf like Basra would be granting an enormous logistical advantage to a strategic competitor, particularly with the difficult relationship with adjacent Iran. Similarly, the importance of the Kurds to the internal issues within Turkey, a vital NATO partner and strategic ally against Russian ambition, would rule out granting Moscow significant influence in the north of Iraq.
So US politics (oil-industry backing for the GOP) is against giving Russia the opportunity to exploit Iraqi oil, and US foreign policy is against giving Russia the influence that would come with the opportunity to exploit Iraqi oil, so unless there is some compelling further reason why the whole campaign could not go ahead without Russian backing, I can't see it working out.