Well first, the chronology here is messed up, the white movement were not Monarchists, their roots as a group lie in the February revolution. Even the most conservative of them had no wish to restore the monarchy and were mostly happy with a military dictatorship.
Is it easy to tell? Was there not a doctrine of deliberately avoiding making statements on how Russia would be governed after the war was won, in order to avoid splitting the White movement?
I doubt the victorious Whites would restore the Romanovs to absolute power, but I can easily imagine a de jure Romanov emperor as a figleaf to legitimise a de facto conservative stratocracy led by whoever emerged from the presumable power struggle between Denikin, Kolchak, Yudenich et al.
Further even if the Bolsheviks are crushed early on in October, the whites would quickly find that Russia had no capacity to continue fighting the war, that the military had essentially totally collapsed, and that the only option was demobilization or facing the fact that what remained of their army wouldn't fight. They'd have to choose Peace or total defeat to Germany.
I would note that the OP supposes an earlier Triple Entente victory, presumably due to changes on the Western Front. I would also note that the Bolsheviks continued the war beyond 1917 in OTL and may well have attempted to continue it longer if Imperial Germany's situation in the west looked sufficiently precarious. Indeed, I suggested this as a way to weaken public support of the Bolsheviks, on the basis that a cry for peace and not imperialist war looks less credible when the movement making it is continuing the very war that it is decrying as imperialist; that way, if the Whites were also given more support by the West than they were in OTL and if the Whites got sufficiently lucky, the Whites could potentially have seized the poor areas of central Russia that were the key source of Bolshevik support.
In stodge's way of putting it, the way I interpreted the OP was "What if the Bolshevik Government was overthrown during the Russian Civil War?" rather than supposing an earlier Bolshevik collapse.