The best way to finish the Abyss is to take out the Demon Lords but leave the Devil and Yugoloth lords in place so they can invade. But I'm not even sure that's enough.
Yeah... the infinities involved do not have clearly defined orders to them (each layer of the nine hells are defined as infinite... but the semi-defined 'infinite' layers of the abyss (not necessarily infinite, though, as it's said there are 666 of them... which may or may not be accurate) are neither specifically all finite, nor specifically all infinite.
Let's make an assumption:
The Blood War has not been resolved because the 'resources' of the Demons and the Devils are approximately equal-sized infinities for every reasonable and useful definition of 'resources'
And because the OP doen't seem interested in them, let's make another assumption:
The Yugoloths don't have an impact in this analysis
And another:
There are a finite number of demon lords and devil lords
And a final:
Any given Demon Lord or Devil Lord has a finite amount of 'resources' that go away when the respective Lord is killed
Assuming those assumptions (redundant redundancy is redundant for the sake of emphatic emphasis), then killing all the demon and devil lords would not have an appreciable impact on the situation
as a whole as it reduces the resources of two infinities by a finite amount, doesn't really change the goals, and will soon settle out with new Demon/Devil lords (via succession, subschool Bloody, descriptor VERY).
That said, the short-term turmoil would be very useful for individuals with goals. Why? Because for a short period of time, there's no new orders coming down from the top to adapt to variables for the orderly devils, nor to reign-in chaos for the disorderly demons - so for the Devils, defensive patrols will be more predictible, and for the Demons, strategic assets will be less densely defended. Anyone wanting to get something specific will be more able to do so.